Powers and Roots

Introduction

Powers (also called indices or exponents) and roots are two sides of the same coin: a power compresses repeated multiplication into a compact form, while a root “undoes” that power. For example, \(2^5\) means \(2 \times 2 \times 2 \times 2 \times 2 = 32\), whereas \(\sqrt{49}\) asks “which number squared gives 49?”, and the answer is 7. This simple idea scales up to many GCSE topics: index laws, standard form (powers of ten), surds (exact square roots that can’t be written as whole numbers), and solving equations that involve squared or cubed quantities.

Why is this important? First, because powers and roots appear everywhere in the GCSE syllabus: Number (indices and surds), Algebra (quadratics and completing the square), and even Geometry and Measures (area scales with the square of length, volume with the cube). Second, because they model real phenomena. Exponential growth and decay describe population change and radioactive half-life; inverse-square relationships appear in physics (e.g. light intensity and gravitational force); and powers of ten are the language of modern science and computing (from nanometres to gigabytes).

Here are a few places you will see powers and roots in action:

• Area and volume scaling: If the side length of a square doubles, the area becomes \(2^2 = 4\) times larger. If the edge of a cube triples, the volume becomes \(3^3 = 27\) times larger.
• Scientific notation: Astronomical and microscopic quantities are written with powers of ten, e.g. \(6.02 \times 10^{23}\) or \(3.0 \times 10^{-9}\).
• Physics and engineering: Energy stored in a spring is proportional to the square of extension; sound intensity and earthquake energy are compared on logarithmic (power-based) scales.
• Computing: Algorithm complexities such as \(O(n^2)\) and \(O(2^n)\) are expressed with powers, capturing how quickly running time grows as inputs increase.
• Finance: Compound interest grows by a power each period, e.g. \(A = P(1 + r)^n\).

At GCSE level, you’ll build fluency with the index laws—rules for simplifying products and quotients of powers with the same base—then extend those ideas to zero, negative, and fractional indices. Understanding fractional indices is especially powerful: it links roots and exponents seamlessly, e.g. \(\sqrt[3]{8} = 8^{1/3} = 2\). You’ll also work with surds, keeping exact values like \(\sqrt{3}\) in simplest form and learning to rationalise denominators so that square roots do not appear on the bottom of a fraction.

A few quick “sense checks” motivate careful thinking:

• Magnitude check: If \(0 < a < 1\), then \(a^2\) is smaller than \(a\) (e.g. \(0.2^2 = 0.04\)). If \(a > 1\), \(a^2\) is larger than \(a\) (e.g. \(5^2 = 25\)).
• Sign check: Squaring a negative number yields a positive result (e.g. \((-3)^2 = 9\)), but odd powers preserve the sign (e.g. \((-3)^3 = -27\)).
• Bracket awareness: \(-3^2\) means \(-(3^2) = -9\) while \((-3)^2 = 9\). Brackets change the meaning and the answer.

This tutorial is designed to take you from the core ideas to confident application: you will learn the index laws, practise converting between roots and powers, simplify and combine surds, and rationalise denominators clearly and efficiently. Each new rule comes with worked examples, common pitfalls, and practice questions that range from straightforward to challenging. By the end, you should be able to handle expressions like \( \dfrac{4a^{-2}\sqrt{18}}{2\sqrt{2}} \), simplify them to exact form, and check your results using estimation or substitution.

If you’re already comfortable with multiplication, division, fractions, and factorising integers, you’re ready to begin. If not, a quick revision of times tables, prime factorisation, and fraction arithmetic will make everything that follows much smoother. Either way, this is the moment where indices stop being mysterious “little numbers” and become a clean, powerful language for describing growth, scale, and structure.

Key Vocabulary

To work confidently with powers and roots, you need to know the precise meaning of the words and symbols that appear in questions. Examiners expect you to use the correct vocabulary and to understand what each instruction implies. Here are the essential terms for this topic:

• Base: The number being raised to a power. In \(5^3\), the base is 5.
• Power / Index / Exponent: The small number written above and to the right of the base. It tells you how many times the base is multiplied by itself. In \(5^3\), the power is 3, meaning \(5 \times 5 \times 5 = 125\).
• Square: Raising a number to the power of 2. Example: \(9^2 = 81\). We often say “9 squared”.
• Cube: Raising a number to the power of 3. Example: \(4^3 = 64\). We often say “4 cubed”.
• Root: The inverse of raising to a power. Example: \(\sqrt{81} = 9\) because \(9^2 = 81\). Similarly, \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
• Square root: A number that, when squared, gives the original number. Example: \(\sqrt{36} = 6\).
• Cube root: A number that, when cubed, gives the original number. Example: \(\sqrt[3]{64} = 4\).
• Surd: An exact root that cannot be simplified into a whole number or terminating decimal. Example: \(\sqrt{2}\) and \(\sqrt{5}\) are surds.
• Rational number: Any number that can be written as a fraction \(\tfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Example: \(\tfrac{3}{4}, 0.25, -2\).
• Irrational number: A number that cannot be written exactly as a fraction. Its decimal expansion is infinite and non-repeating. Examples: \(\pi, e, \sqrt{2}\).
• Index laws: Rules that govern how to simplify expressions involving powers with the same base. Example: \(a^m \times a^n = a^{m+n}\).
• Rationalising: The process of rewriting a fraction so that the denominator contains no surds. Example: \(\tfrac{1}{\sqrt{3}} = \tfrac{\sqrt{3}}{3}\).

These terms will appear repeatedly throughout the tutorial. Make sure you are comfortable with them before moving on, as they form the essential “language of powers and roots”.

The Index Laws

The index laws (sometimes called laws of indices) are the rules that tell us how to simplify expressions involving powers with the same base. They allow you to combine, split, or manipulate powers systematically without having to expand everything out. For GCSE Mathematics, you need to be fluent with each of these rules and know when to apply them.

1) Multiplying Powers with the Same Base

If the base is the same, add the exponents: \[ a^m \times a^n = a^{m+n}. \] This works because multiplication is repeated addition of exponents.

• Example A: \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\).
• Example B: \(x^5 \times x^2 = x^{5+2} = x^7\).

2) Dividing Powers with the Same Base

If the base is the same, subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0. \]

• Example C: \(\tfrac{5^7}{5^3} = 5^{7-3} = 5^4 = 625\).
• Example D: \(\tfrac{x^9}{x^4} = x^{9-4} = x^5\).

3) Power of a Power

When raising a power to another power, multiply the exponents: \[ (a^m)^n = a^{m \times n}. \]

• Example E: \((3^2)^4 = 3^{2 \times 4} = 3^8 = 6561\).
• Example F: \((x^5)^3 = x^{15}\).

4) Power of a Product

If two factors are multiplied, each gets the exponent: \[ (ab)^m = a^m b^m. \]

• Example G: \((2 \times 3)^4 = 2^4 \times 3^4 = 16 \times 81 = 1296\).
• Example H: \((xy)^3 = x^3 y^3\).

5) Power of a Quotient

If a fraction is raised to a power, the power applies to numerator and denominator: \[ \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, \quad b \neq 0. \]

• Example I: \(\left(\tfrac{3}{4}\right)^2 = \tfrac{3^2}{4^2} = \tfrac{9}{16}\).
• Example J: \(\left(\tfrac{x}{y}\right)^3 = \tfrac{x^3}{y^3}\).

Worked Examples (Mixed)

• Example K: Simplify \(x^4 \times x^7\). Solution: \(= x^{4+7} = x^{11}\).
• Example L: Simplify \(\frac{y^{12}}{y^5}\). Solution: \(= y^{12-5} = y^7\).
• Example M: Simplify \((2^3)^5\). Solution: \(= 2^{15} = 32,768\).
• Example N: Simplify \((3xy)^2\). Solution: \(= 3^2 \times x^2 \times y^2 = 9x^2y^2\).
• Example O: Simplify \(\left(\tfrac{4}{5}\right)^3\). Solution: \(= \tfrac{4^3}{5^3} = \tfrac{64}{125}\).

Mastering these five rules means you can simplify almost any expression involving integer powers. In the next section, we’ll extend these rules to special exponents — zero, negative, and fractional powers — which make the system of indices complete.

Special Exponents

So far, we have worked with positive integer powers like \(2^3 = 8\). But what happens if the exponent is zero, negative, or a fraction? These cases are not new rules — they are natural extensions of the index laws you have already seen. Understanding them allows us to work with a complete system of powers that includes reciprocals, roots, and much more.

1) Zero Index

Any non-zero number raised to the power of zero equals 1: \[ a^0 = 1, \quad a \neq 0. \] This follows directly from the division rule. For example: \[ \frac{a^5}{a^5} = a^{5-5} = a^0. \] But \(\frac{a^5}{a^5} = 1\), so \(a^0 = 1\).

• Example A: \(7^0 = 1\).
• Example B: \((-3)^0 = 1\).

Be careful: \(0^0\) is undefined, so it will never appear in GCSE questions.

2) Negative Indices

A negative index means “take the reciprocal”. In other words: \[ a^{-n} = \frac{1}{a^n}, \quad a \neq 0. \] This rule also comes from the division law. For example: \[ \frac{a^3}{a^5} = a^{3-5} = a^{-2} = \frac{1}{a^2}. \]

• Example C: \(2^{-3} = \tfrac{1}{2^3} = \tfrac{1}{8}\).
• Example D: \(10^{-1} = \tfrac{1}{10} = 0.1\).
• Example E: \(x^{-4} = \tfrac{1}{x^4}\).

3) Fractional Indices

A fractional index represents a root. The denominator of the fraction gives the root, and the numerator gives the power: \[ a^{\tfrac{1}{n}} = \sqrt[n]{a}, \quad a^{\tfrac{m}{n}} = \sqrt[n]{a^m}. \]

• Example F: \(9^{1/2} = \sqrt{9} = 3\).
• Example G: \(27^{1/3} = \sqrt[3]{27} = 3\).
• Example H: \(16^{3/4} = \sqrt[4]{16^3} = \sqrt[4]{4096} = 8\).

Fractional powers let us rewrite roots as exponents, so that all index laws continue to apply. For example: \[ \sqrt{a} \times \sqrt{a} = a^{1/2} \times a^{1/2} = a^{1/2 + 1/2} = a^1 = a. \]

4) Combining Negative and Fractional Indices

A negative fractional index means both “reciprocal” and “root”. For example: \[ a^{-1/2} = \frac{1}{a^{1/2}} = \frac{1}{\sqrt{a}}. \]

• Example I: \(25^{-1/2} = \tfrac{1}{25^{1/2}} = \tfrac{1}{5}\).
• Example J: \(8^{-2/3} = \tfrac{1}{8^{2/3}} = \tfrac{1}{(\sqrt[3]{8})^2} = \tfrac{1}{4}\).

Worked Examples (Special Exponents)

• Example K: Simplify \(12^0\). Solution: \(= 1\).
• Example L: Simplify \(4^{-2}\). Solution: \(= \tfrac{1}{4^2} = \tfrac{1}{16}\).
• Example M: Simplify \(64^{1/2}\). Solution: \(= \sqrt{64} = 8\).
• Example N: Simplify \(81^{-3/4}\). Solution: \(= \tfrac{1}{81^{3/4}} = \tfrac{1}{(\sqrt[4]{81})^3} = \tfrac{1}{3^3} = \tfrac{1}{27}\).

These extensions make the system of indices complete. With zero, negative, and fractional powers, we can handle reciprocals, roots, and combinations all within the same framework. This means the index laws never need exceptions — they always work if you interpret the powers correctly.

Roots as Powers

We often meet roots written with the radical symbol \(\sqrt{}\), but it is very useful to rewrite them as fractional powers. Doing so allows us to apply the index laws seamlessly, rather than having one set of rules for powers and another set for roots.

1) Square Roots

The square root of \(a\) is the number which, when squared, gives \(a\). In power form: \[ \sqrt{a} = a^{1/2}. \]

• Example A: \(\sqrt{49} = 49^{1/2} = 7\).
• Example B: \(\sqrt{x^2} = (x^2)^{1/2} = x^{2 \times 1/2} = x^1 = x\) (for positive \(x\)).

2) Cube Roots

The cube root of \(a\) is the number which, when cubed, gives \(a\). In power form: \[ \sqrt[3]{a} = a^{1/3}. \]

• Example C: \(\sqrt[3]{27} = 27^{1/3} = 3\).
• Example D: \(\sqrt[3]{x^6} = (x^6)^{1/3} = x^{6/3} = x^2\).

3) Higher Roots

The \(n\)th root of \(a\) can be written as: \[ \sqrt[n]{a} = a^{1/n}. \]

• Example E: \(\sqrt[4]{16} = 16^{1/4} = 2\).
• Example F: \(\sqrt[5]{32} = 32^{1/5} = 2\).

4) Roots with Additional Powers

If the power is a fraction \(\tfrac{m}{n}\), the denominator gives the root and the numerator gives the power: \[ a^{m/n} = \sqrt[n]{a^m} = \big(\sqrt[n]{a}\big)^m. \] Both interpretations give the same result.

• Example G: \(27^{2/3} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9\).
• Example H: \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\).

5) Using Index Laws with Roots

The great advantage of writing roots as powers is that the index laws apply without change.

• Example I: \(\sqrt{a} \times \sqrt{a} = a^{1/2} \times a^{1/2} = a^{1} = a\).
• Example J: \(\frac{\sqrt[3]{x^5}}{\sqrt[3]{x^2}} = x^{5/3 - 2/3} = x^{3/3} = x\).
• Example K: \((\sqrt[4]{y})^8 = (y^{1/4})^8 = y^{8/4} = y^2\).

By recognising roots as powers, we unify two major ideas in mathematics. Instead of having separate sets of rules for exponents and roots, we have one consistent system — the index laws — that works for all positive integers, negative exponents, and fractions alike.

Order of Operations with Powers & Roots

Whenever you simplify or evaluate an expression with powers and roots, the order of operations matters. This is usually remembered with the acronym BIDMAS (Brackets, Indices, Division/Multiplication, Addition, Subtraction). Powers and roots fall under the “Indices” step, which comes before multiplying, dividing, adding, or subtracting.

1) BIDMAS Reminders

• Brackets: Simplify inside brackets first.
• Indices: Apply powers and roots next.
• Division/Multiplication: Work left to right.
• Addition/Subtraction: Work left to right last.

This order avoids ambiguity and ensures everyone gets the same result.

2) Examples with Powers and Roots

• Example A: Simplify \(2 + 3^2\). Apply indices first: \(3^2 = 9\). Then add: \(2 + 9 = 11\).
• Example B: Simplify \((2 + 3)^2\). Brackets first: \(2 + 3 = 5\). Then square: \(5^2 = 25\). Note: This is not the same as \(2 + 3^2\)!
• Example C: Simplify \(16 \div 2^2\). Indices first: \(2^2 = 4\). Then divide: \(16 \div 4 = 4\).
• Example D: Simplify \(\sqrt{36} + 2^3\). Roots and indices first: \(\sqrt{36} = 6\), \(2^3 = 8\). Then add: \(6 + 8 = 14\).

3) Nested Powers and Roots

Sometimes powers and roots are nested inside each other. Work from the inside out, applying BIDMAS carefully.

• Example E: Simplify \(\sqrt{3^2 + 7}\). Inside: \(3^2 = 9\). Then add: \(9 + 7 = 16\). Finally, root: \(\sqrt{16} = 4\).
• Example F: Simplify \((\sqrt{25})^3\). Root first: \(\sqrt{25} = 5\). Then cube: \(5^3 = 125\).
• Example G: Simplify \(\sqrt[3]{2^4}\). Power first: \(2^4 = 16\). Then cube root: \(\sqrt[3]{16}\). This is not a whole number, so leave it as \(\sqrt[3]{16}\) or approximate with a calculator.

4) Common Pitfalls

• Misusing brackets: \(-3^2\) means \(-(3^2) = -9\), but \((-3)^2 = 9\). Always check where brackets are placed.
• Forgetting BIDMAS: In \(2 + 3^2\), some students mistakenly do \(2 + 3 = 5\), then square, giving 25. The correct answer is 11.
• Mixing root and power order: In \(\sqrt{4^2}\), apply the power first: \(4^2 = 16\), then root: \(\sqrt{16} = 4\).

By mastering the order of operations with powers and roots, you avoid the most common exam mistakes and ensure your answers are accurate and consistent.

Squares, Cubes & Higher Powers

Squares and cubes are the most common powers that GCSE students work with. Knowing them quickly helps with mental maths, estimation, and spotting patterns in algebra. Higher powers (like fourth or fifth powers) also appear occasionally, especially in surds and fractional indices.

1) Square Numbers

A square number is the result of an integer multiplied by itself. You should memorise square numbers at least up to \(15^2 = 225\), as they appear frequently in non-calculator papers.

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)
• \(6^2 = 36\)
• \(7^2 = 49\)
• \(8^2 = 64\)
• \(9^2 = 81\)
• \(10^2 = 100\)
• \(11^2 = 121\)
• \(12^2 = 144\)
• \(13^2 = 169\)
• \(14^2 = 196\)
• \(15^2 = 225\)

Beyond 15, you should still know a few key values: \(20^2 = 400\), \(25^2 = 625\), \(30^2 = 900\), and \(50^2 = 2500\). These are useful benchmarks for estimation.

2) Cube Numbers

A cube number is the result of an integer multiplied by itself three times. You should know cubes up to \(10^3 = 1000\).

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• \(6^3 = 216\)
• \(7^3 = 343\)
• \(8^3 = 512\)
• \(9^3 = 729\)
• \(10^3 = 1000\)

It is also worth remembering \(12^3 = 1728\) and \(15^3 = 3375\), as they appear in more challenging problems.

3) Fourth and Higher Powers

While less common, you may meet fourth powers or higher in fractional index problems. Knowing how to build them quickly is helpful:

• \(2^4 = 16\)
• \(3^4 = 81\)
• \(5^4 = 625\)
• \(10^4 = 10,000\)

You don’t need to memorise higher powers beyond this, but you should recognise that they grow rapidly: \(2^{10} = 1024\), which is already above one thousand.

4) Perfect Squares and Cubes

Numbers like 36, 49, 64, 81, 100 are perfect squares. Numbers like 27, 64, 125, 216 are perfect cubes. Being able to recognise these instantly allows you to simplify square roots and cube roots efficiently:

• \(\sqrt{64} = 8\) because 64 is a perfect square.
• \(\sqrt[3]{216} = 6\) because 216 is a perfect cube.

5) Applications in GCSE Maths

• Area: The area of a square with side length \(a\) is \(a^2\).
• Volume: The volume of a cube with side length \(a\) is \(a^3\).
• Quadratics: Expanding \((x+3)^2\) always gives a square term \(x^2\).
• Standard form: Scientific notation relies heavily on powers of 10, e.g. \(10^6\), \(10^{-9}\).

By memorising square and cube numbers, and recognising perfect powers quickly, you will save valuable time in exams and reduce careless errors.

Square & Cube Roots

Roots are the inverse of powers. The square root of a number is the value that, when squared, gives the original number. The cube root of a number is the value that, when cubed, gives the original number. At GCSE level, square and cube roots are the most common, but you may also encounter higher roots in fractional indices.

1) Square Roots

The square root of \(a\) is written \(\sqrt{a}\). If \(b^2 = a\), then \(b = \sqrt{a}\). For example:

• \(\sqrt{25} = 5\) because \(5^2 = 25\).
• \(\sqrt{81} = 9\) because \(9^2 = 81\).
• \(\sqrt{2}\) has no exact whole number result, so it is left in root (surd) form.

Note: \(\sqrt{a}\) always refers to the principal (positive) root. Although both \(+7\) and \(-7\) satisfy \((\pm7)^2 = 49\), by convention \(\sqrt{49} = 7\).

2) Cube Roots

The cube root of \(a\) is written \(\sqrt[3]{a}\). If \(b^3 = a\), then \(b = \sqrt[3]{a}\). Examples:

• \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
• \(\sqrt[3]{-64} = -4\) because \((-4)^3 = -64\).
• \(\sqrt[3]{125} = 5\).

Unlike square roots, cube roots can be negative, since odd powers preserve the sign of the base.

3) Prime Factor Method for Square Roots

One way to simplify square roots is to use prime factorisation. Break the number into prime factors, then pair them up:

• Example A: Simplify \(\sqrt{72}\). Factorise: \(72 = 2 \times 2 \times 2 \times 3 \times 3\). Pairing: take one 2 and one 3 out of the root. Result: \(\sqrt{72} = 6\sqrt{2}\).
• Example B: Simplify \(\sqrt{200}\). Factorise: \(200 = 2 \times 2 \times 2 \times 5 \times 5\). Pairs: a 2 and a 5 come out. Result: \(\sqrt{200} = 10\sqrt{2}\).

4) Estimating Roots

When roots are not perfect squares or cubes, you can estimate them by comparing with known values:

• \(\sqrt{50}\) is between \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\), so \(\sqrt{50} \approx 7.07\).
• \(\sqrt[3]{30}\) is between \(\sqrt[3]{27} = 3\) and \(\sqrt[3]{64} = 4\), so \(\sqrt[3]{30} \approx 3.11\).

5) Using a Calculator

On calculator papers, you can find square roots using the \(\sqrt{}\) button, and cube roots or higher roots using the \(x^{1/n}\) or \(n\sqrt{}\) functions. In exams, exact surd form (like \(\sqrt{2}\)) is often preferred over decimals, unless the question explicitly asks for an approximation.

By mastering square and cube roots, you prepare yourself for simplifying surds, solving quadratic equations, and working with fractional indices later in this tutorial.

Surds — Basics

A surd is a root that cannot be simplified into a whole number or a terminating/repeating decimal. They are exact values that stay in root form. For example:

• \(\sqrt{2}, \sqrt{3}, \sqrt{5}\) are surds — they cannot be simplified.
• \(\sqrt{4} = 2\) is not a surd, because it simplifies to a whole number.

Why use surds? Because they preserve accuracy. If you approximate \(\sqrt{2}\) as 1.414, you introduce rounding error. But if you keep it as \(\sqrt{2}\), the answer stays exact until the very end of a calculation.

1) Identifying Surds

Surds usually appear in problems involving Pythagoras’ Theorem, trigonometry, and area/volume calculations. You can spot a surd if:

• The root is not a perfect square or cube (e.g. \(\sqrt{20}\)).
• The decimal result is infinite and non-repeating (e.g. \(\sqrt{3} \approx 1.732...\)).

2) Simplifying Surds

To simplify a surd, look for square factors inside the root: \[ \sqrt{ab} = \sqrt{a} \times \sqrt{b}. \] Take the square factors out of the root, leave the rest inside.

• Example A: \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\).
• Example B: \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\).
• Example C: \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).

3) Like and Unlike Surds

You can only add or subtract surds if they are “like” surds (same root). Otherwise, leave them separate.

• \(2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}\).
• \(\sqrt{2} + \sqrt{5}\) cannot be simplified further.

4) Multiplying Surds

Multiplication works neatly with surds because you can combine them under one root: \[ \sqrt{a} \times \sqrt{b} = \sqrt{ab}. \]

• Example D: \(\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4\).
• Example E: \(3\sqrt{5} \times 2\sqrt{3} = 6\sqrt{15}\).

5) Why Keep Surds?

Surds are kept in exact form in GCSE exams because:

• They avoid rounding errors in multi-step calculations.
• They make algebra cleaner (e.g. \(\frac{1}{\sqrt{2}}\) rationalises to \(\frac{\sqrt{2}}{2}\), a standard form in trigonometry).
• They show full mathematical understanding rather than relying on decimals.

Understanding the basics of surds sets the stage for the next steps: combining them through addition, subtraction, multiplication, and rationalising denominators.

Simplifying Surds

Simplifying surds means rewriting them in their neatest form, with no square factors left inside the root. This process is important in GCSE exams because it shows full mathematical reasoning and makes later steps (like combining terms or rationalising) much easier.

1) The Rule for Simplifying

Look for perfect square factors inside the root: \[ \sqrt{ab} = \sqrt{a} \times \sqrt{b}. \] If one of the factors is a square number, it can come out of the root.

2) Examples

• Example A: \(\sqrt{12}\). Factorise: \(12 = 4 \times 3\). Simplify: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3}\).
• Example B: \(\sqrt{72}\). Factorise: \(72 = 36 \times 2\). Simplify: \(\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}\).
• Example C: \(\sqrt{98}\). Factorise: \(98 = 49 \times 2\). Simplify: \(\sqrt{98} = \sqrt{49}\sqrt{2} = 7\sqrt{2}\).
• Example D: \(\sqrt{200}\). Factorise: \(200 = 100 \times 2\). Simplify: \(\sqrt{200} = \sqrt{100}\sqrt{2} = 10\sqrt{2}\).

3) General Method

1. Factorise the number under the root.
2. Identify the largest perfect square factor.
3. Take the square root of that factor out in front.
4. Leave the rest inside the root.

4) Surds with Coefficients

Sometimes surds already have a number in front. Multiply that by whatever comes out of the root:

• Example E: \(3\sqrt{20}\). Simplify: \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\). Multiply: \(3 \times 2\sqrt{5} = 6\sqrt{5}\).
• Example F: \(5\sqrt{72}\). Simplify: \(\sqrt{72} = 6\sqrt{2}\). Multiply: \(5 \times 6\sqrt{2} = 30\sqrt{2}\).

5) Cube Roots and Higher

The same idea works for cube roots (and higher). Look for cube factors:

• Example G: \(\sqrt[3]{54}\). Factorise: \(54 = 27 \times 2\). Simplify: \(\sqrt[3]{54} = \sqrt[3]{27}\sqrt[3]{2} = 3\sqrt[3]{2}\).
• Example H: \(\sqrt[3]{250}\). Factorise: \(250 = 125 \times 2\). Simplify: \(\sqrt[3]{250} = \sqrt[3]{125}\sqrt[3]{2} = 5\sqrt[3]{2}\).

6) Why Simplify?

• It makes answers shorter and cleaner.
• It helps when combining like terms (e.g. \(2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}\)).
• It avoids losing marks in exams where “simplify your answer” is required.

Simplifying surds is the first step in working with them confidently. Next, we will look at adding, subtracting, and multiplying surds, and how these rules mirror those for algebraic terms.

Adding, Subtracting & Multiplying Surds

Once surds are simplified, you can combine them in similar ways to algebraic terms. Think of each surd like a “variable” — you can only add or subtract like terms, but multiplication and distribution follow the same rules as expanding brackets.

1) Adding Surds

You can only add surds if they are the same type (i.e. the same number inside the root). These are called like surds.

• \(\sqrt{2} + \sqrt{2} = 2\sqrt{2}\).
• \(3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}\).
• \(\sqrt{3} + \sqrt{7}\) cannot be simplified further because the surds are unlike.

2) Subtracting Surds

Subtracting follows the same principle — only like surds can be combined.

• \(6\sqrt{2} - 4\sqrt{2} = 2\sqrt{2}\).
• \(5\sqrt{7} - \sqrt{7} = 4\sqrt{7}\).
• \(2\sqrt{3} - \sqrt{5}\) cannot be simplified because the surds are different.

3) Multiplying Surds

Multiplication works across all surds, even unlike ones, because you can combine them under a single root: \[ \sqrt{a} \times \sqrt{b} = \sqrt{ab}. \]

• Example A: \(\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6\).
• Example B: \(2\sqrt{5} \times 3\sqrt{2} = (2 \times 3)(\sqrt{5} \times \sqrt{2}) = 6\sqrt{10}\).
• Example C: \(\sqrt{7} \times \sqrt{7} = \sqrt{49} = 7\).

4) Expanding with Surds

When multiplying brackets containing surds, expand them as you would with algebra.

• Example D: \((\sqrt{2} + 3)(\sqrt{2} + 5)\). Expand: \(\sqrt{2} \cdot \sqrt{2} + 5\sqrt{2} + 3\sqrt{2} + 15\). Simplify: \(2 + 8\sqrt{2} + 15 = 17 + 8\sqrt{2}\).
• Example E: \((\sqrt{3} - 1)(\sqrt{3} + 1)\). Expand: \(\sqrt{3}\cdot\sqrt{3} + \sqrt{3}\cdot 1 - 1\cdot\sqrt{3} - 1\cdot 1\). Simplify: \(3 + \sqrt{3} - \sqrt{3} - 1 = 2\). (This is a difference of squares pattern.)

5) Dividing Surds (Pre-Rationalising)

Division is similar to multiplication: combine under one root if possible.

• Example F: \(\tfrac{\sqrt{18}}{\sqrt{2}} = \sqrt{\tfrac{18}{2}} = \sqrt{9} = 3\).
• Example G: \(\tfrac{6\sqrt{5}}{2\sqrt{5}} = \tfrac{6}{2} \times \tfrac{\sqrt{5}}{\sqrt{5}} = 3\).

With these rules in place, we are ready to tackle a key GCSE requirement: rationalising denominators.

Rationalising Denominators

In mathematics, it is considered poor form to leave a surd in the denominator of a fraction. Instead, we rationalise the denominator, meaning we rewrite the fraction so that the denominator is a rational number (no roots left). This makes expressions cleaner and easier to use in further calculations.

1) Single Surd in the Denominator

If the denominator is a single square root, multiply top and bottom by the same root. This uses the fact that \(\sqrt{a} \times \sqrt{a} = a\).

• Example A: \(\tfrac{1}{\sqrt{5}}\). Multiply by \(\tfrac{\sqrt{5}}{\sqrt{5}}\): \(\tfrac{1}{\sqrt{5}} \times \tfrac{\sqrt{5}}{\sqrt{5}} = \tfrac{\sqrt{5}}{5}\).
• Example B: \(\tfrac{7}{\sqrt{3}}\). Multiply by \(\tfrac{\sqrt{3}}{\sqrt{3}}\): \(= \tfrac{7\sqrt{3}}{3}\).

2) Denominator with a Coefficient and Surd

If the denominator is of the form \(k\sqrt{a}\), multiply top and bottom by \(\sqrt{a}\).

• Example C: \(\tfrac{2}{3\sqrt{2}}\). Multiply by \(\tfrac{\sqrt{2}}{\sqrt{2}}\): \(= \tfrac{2\sqrt{2}}{3 \times 2} = \tfrac{\sqrt{2}}{3}\).

3) Binomial Denominator (Two Terms)

If the denominator has two terms, like \(a + \sqrt{b}\), multiply top and bottom by its conjugate \(a - \sqrt{b}\). The difference of squares eliminates the surd.

• Example D: \(\tfrac{1}{2 + \sqrt{3}}\). Multiply by \(\tfrac{2 - \sqrt{3}}{2 - \sqrt{3}}\): Denominator: \((2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1\). Numerator: \(1 \times (2 - \sqrt{3}) = 2 - \sqrt{3}\). Final: \(\tfrac{1}{2 + \sqrt{3}} = 2 - \sqrt{3}\).
• Example E: \(\tfrac{5}{3 - \sqrt{2}}\). Multiply by \(\tfrac{3 + \sqrt{2}}{3 + \sqrt{2}}\): Denominator: \(9 - 2 = 7\). Numerator: \(5(3 + \sqrt{2}) = 15 + 5\sqrt{2}\). Final: \(\tfrac{15 + 5\sqrt{2}}{7}\).

4) General Strategy

1. If there is one surd in the denominator → multiply by that surd.
2. If there are two terms → multiply by the conjugate.
3. Simplify numerator and denominator fully.

5) Why Rationalise?

• It avoids surds in denominators, which is standard mathematical form.
• It makes comparing fractions easier (common denominators become possible).
• It ensures exact answers in trigonometry, geometry, and algebra.

Rationalising denominators is one of the key Higher GCSE skills with surds. It is frequently tested in exam papers, often as part of a multi-step question. Mastering it ensures that your answers are always in the expected, fully simplified form.

Powers & Roots with Fractions and Decimals

So far we have worked mainly with whole numbers, but GCSE questions often involve fractions and decimals as bases. The same index laws apply — powers and roots behave consistently regardless of the type of number. You just need to take extra care with calculations and simplifications.

1) Powers of Fractions

When a fraction is raised to a power, raise both numerator and denominator: \[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}. \]

• Example A: \(\left(\tfrac{2}{3}\right)^2 = \tfrac{2^2}{3^2} = \tfrac{4}{9}\).
• Example B: \(\left(\tfrac{3}{5}\right)^3 = \tfrac{3^3}{5^3} = \tfrac{27}{125}\).

2) Roots of Fractions

Roots apply to numerator and denominator separately: \[ \sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}. \]

• Example C: \(\sqrt{\tfrac{9}{16}} = \tfrac{\sqrt{9}}{\sqrt{16}} = \tfrac{3}{4}\).
• Example D: \(\sqrt[3]{\tfrac{8}{27}} = \tfrac{\sqrt[3]{8}}{\sqrt[3]{27}} = \tfrac{2}{3}\).

3) Negative and Fractional Indices with Fractions

Fractional indices work the same way with fractions: \[ \left(\tfrac{a}{b}\right)^{m/n} = \frac{a^{m/n}}{b^{m/n}}. \]

• Example E: \(\left(\tfrac{4}{9}\right)^{1/2} = \tfrac{2}{3}\).
• Example F: \(\left(\tfrac{27}{8}\right)^{-2/3} = \left(\tfrac{27^{1/3}}{8^{1/3}}\right)^{-2} = \left(\tfrac{3}{2}\right)^{-2} = \left(\tfrac{2}{3}\right)^2 = \tfrac{4}{9}\).

4) Powers of Decimals

Decimals can also be raised to powers, but the results are often smaller:

• Example G: \(0.2^2 = 0.04\).
• Example H: \(0.5^3 = 0.125\).
• Example I: \(1.2^4 = 2.0736\).

5) Roots of Decimals

Square and cube roots of decimals are usually approximations, but can sometimes be neat:

• Example J: \(\sqrt{0.25} = 0.5\).
• Example K: \(\sqrt{0.04} = 0.2\).
• Example L: \(\sqrt[3]{0.008} = 0.2\).

6) Common Traps

• Be careful with negative indices on fractions: \(\left(\tfrac{2}{3}\right)^{-1} = \tfrac{3}{2}\), not \(\tfrac{2}{3}\).
• Don’t confuse \(\sqrt{\tfrac{a}{b}}\) with \(\tfrac{\sqrt{a+b}}{}\). The square root applies to numerator and denominator separately, not to the sum.
• Watch out for rounding errors — leave answers as exact fractions or surds unless asked for decimals.

Handling fractions and decimals with powers and roots is an essential skill in GCSE. It connects with topics such as ratio, percentages, and standard form, and often appears in exam questions where accuracy and simplification are key.

Scientific Notation Links (Powers of 10)

Scientific notation (also called standard form in GCSE) is a way of writing very large or very small numbers using powers of 10. It relies directly on the rules of indices, so mastering powers makes scientific notation much easier.

1) Standard Form Definition

A number is in standard form if it is written as: \[ a \times 10^n, \] where \(1 \leq a < 10\) and \(n\) is an integer.

• \(3.6 \times 10^5 = 360,000\).
• \(7.1 \times 10^{-3} = 0.0071\).

2) Large Numbers (Positive Powers)

Powers of 10 shift the decimal point to the right.

• Example A: \(2.5 \times 10^4 = 25,000\).
• Example B: \(9.81 \times 10^6 = 9,810,000\).

3) Small Numbers (Negative Powers)

Negative powers of 10 shift the decimal point to the left.

• Example C: \(4.2 \times 10^{-2} = 0.042\).
• Example D: \(7.5 \times 10^{-5} = 0.000075\).

4) Converting to Standard Form

1. Move the decimal point so the first number is between 1 and 9.
2. Count how many places you moved it.
3. If you moved it left → positive power; if you moved it right → negative power.

• Example E: 53,000 → \(5.3 \times 10^4\) (moved 4 places left).
• Example F: 0.00082 → \(8.2 \times 10^{-4}\) (moved 4 places right).

5) Multiplying and Dividing in Standard Form

Use the index laws for powers of 10:

• \((2 \times 10^3) \times (3 \times 10^5) = (2 \times 3) \times 10^{3+5} = 6 \times 10^8\).
• \(\tfrac{6.4 \times 10^7}{8 \times 10^3} = \tfrac{6.4}{8} \times 10^{7-3} = 0.8 \times 10^4 = 8 \times 10^3\).

6) Powers of 10 and Approximation

Powers of 10 allow us to estimate the size of a number quickly:

• The speed of light is about \(3 \times 10^8\) m/s.
• A red blood cell is about \(7 \times 10^{-6}\) m across.
• Avogadro’s number is \(6.02 \times 10^{23}\).

7) Why It Matters

• Science: Standard form is used for measurements in physics, chemistry, and biology.
• Engineering: Tiny components (microchips, nanomaterials) are measured in powers of 10.
• Computing: Storage and processing often scale with powers of 2 and 10.

Scientific notation links powers and roots directly to real-world applications, making it one of the most practical uses of indices in GCSE maths.

Calculator Skills & Exam Mode Tips

While you should know the index laws and simplification rules, efficient use of your calculator is essential in exams. Calculator papers expect you to handle powers, roots, and surds accurately. Here are the key skills and tips.

1) Calculator Buttons

• Powers: Use the x^y or ^ button to raise numbers to any power. Example: enter 2, then x^y, then 5 for \(2^5 = 32\).
• Square: Use the dedicated x² button if available. Example: \(7^2 = 49\).
• Square root: Use the √ button. Example: \(\sqrt{144} = 12\).
• Cubes & higher roots: Use x^y with fractional indices. Example: \(27^{1/3}\) gives the cube root of 27.

2) Exact vs Approximate Values

Your calculator can show decimals, but exams often require exact surd form. Always check what the question asks:

• If it says “give your answer as a surd”, leave \(\sqrt{3}\), not 1.732.
• If it says “give your answer to 2 decimal places”, press = and round appropriately.

3) Storing and Reusing Values

Many problems require multi-step calculations. Rounding too early can lose accuracy. Use the calculator’s ANS key or memory store functions to reuse exact results instead of retyping rounded values.

• Example A: If you calculate \(\sqrt{3}\), store it as a letter (e.g. A) and use \(A^2\) in the next step to keep full accuracy.

4) Checking with Approximation

Estimate first to check whether your calculator answer is reasonable:

• \(\sqrt{50}\) should be just over 7 because \(\sqrt{49} = 7\).
• \(0.2^3\) should be tiny because \(0.2 < 1\).

If your calculator gives an answer like 70 for \(\sqrt{50}\), you know a mistake was made in typing.

5) Rounding & Significant Figures

Exam questions may ask for answers to a given number of decimal places (d.p.) or significant figures (s.f.). Remember:

• Decimal places count digits after the decimal point.
• Significant figures count digits from the first non-zero digit.

• Example B: Round 3.14159 to 2 d.p. → 3.14.
• Example C: Round 0.004562 to 2 s.f. → 0.0046.

6) Common Calculator Mistakes

• Forgetting brackets: entering \(-3^2\) gives -9, not 9. Use \((-3)^2\).
• Mistyping fractional indices: for \(27^{1/3}\), type 27 x^y (1 ÷ 3).
• Rounding too soon: always keep the full calculator value until the end.

Using your calculator effectively is a skill in itself. With good habits — using brackets, storing results, and checking by estimation — you can avoid common traps and maximise your marks.

Quality Checks

Even with good methods, small mistakes can creep into power and root calculations. Quality checks are quick ways to test whether an answer makes sense before moving on. They help you spot sign slips, missing terms, or misplaced decimals.

1) Sign Sense

• Even powers always give positive results: \((-3)^2 = 9\).
• Odd powers preserve the sign: \((-3)^3 = -27\).
• Square roots are always positive by definition: \(\sqrt{25} = 5\), not \(-5\).
• Cube roots can be negative: \(\sqrt[3]{-64} = -4\).

If your answer breaks one of these rules, recheck your calculation.

2) Degree / Power Check

The size of the power should match your expectation:

• \(x^2\) grows faster than \(x\), so \(10^2 = 100\) should be much bigger than 10.
• Fractional powers make numbers smaller (if the base is greater than 1): \(100^{1/2} = 10 < 100\).
• Negative powers give reciprocals: \(10^{-3} = 0.001\).

3) Magnitude Check

Estimate the size of the answer to see if it’s reasonable:

• \(\sqrt{50}\) must be between 7 and 8 because \(7^2 = 49\) and \(8^2 = 64\).
• \(2.5 \times 10^4\) is 25,000, so if your calculator shows 250,000, a decimal shift went wrong.

4) Substitution Spot-Check

When simplifying an algebraic expression involving powers, test your result with a simple value for the variable (like \(x=1\) or \(x=2\)) to see if the original and simplified forms match.

• Example A: Simplify \(x^{3} \times x^{-2} = x\). Check with \(x=2\): LHS \(= 2^3 \times 2^{-2} = 8 \times \tfrac{1}{4} = 2\). RHS \(= 2\). ✓

5) Common-Sense Reality Check

Ask yourself: does the answer fit the context?

• Areas and lengths cannot be negative.
• Roots of small numbers should not give huge values.
• Very large powers (like \(10^9\)) should produce a number with 9 extra zeros.

By applying sign sense, power expectations, magnitude checks, and substitution, you can quickly verify that your work is consistent and accurate before moving on.

Common Mistakes

Even strong students lose marks on powers and roots because of small slips. Here are the most frequent errors and how to avoid them.

1) Mixing Up Negative and Positive Indices

Mistake: Thinking \(2^{-3} = -8\). Correct: \(2^{-3} = \tfrac{1}{2^3} = \tfrac{1}{8}\). Negative powers mean reciprocals, not negative numbers.

2) Forgetting Brackets with Negatives

Mistake: Writing \(-3^2 = 9\). Correct: \(-3^2 = -(3^2) = -9\). But \((-3)^2 = 9\). Always use brackets for negative bases.

3) Confusing Roots and Divisions

Mistake: Treating \(\sqrt{a+b}\) as \(\sqrt{a} + \sqrt{b}\). Correct: \(\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}\). Example: \(\sqrt{9+16} = \sqrt{25} = 5\), but \(\sqrt{9}+\sqrt{16} = 3+4 = 7\).

4) Over-Rounding Surds

Mistake: Writing \(\sqrt{2} = 1.4\) in exact questions. Correct: Leave as \(\sqrt{2}\), unless a decimal approximation is asked for. Premature rounding can cost marks in later steps.

5) Wrong Use of Fractional Indices

Mistake: Thinking \(27^{1/3} = 9\). Correct: \(27^{1/3} = \sqrt[3]{27} = 3\). Fractional powers are roots, not divisions.

6) Misusing Zero Index

Mistake: Believing \(0^0 = 1\) or that \(a^0 = 0\). Correct: For \(a \neq 0\), \(a^0 = 1\). But \(0^0\) is undefined.

7) Forgetting Index Laws

Mistake: \(x^3 \times x^2 = x^5\) is correct, but some students write \(x^6\). Others divide instead of subtracting when simplifying \(\tfrac{x^7}{x^3}\). Correct: Always apply the rules: add powers when multiplying, subtract powers when dividing.

8) Expanding Instead of Squaring

Mistake: Writing \((a+b)^2 = a^2 + b^2\). Correct: \((a+b)^2 = a^2 + 2ab + b^2\). Forgetting the middle term is a very common exam error.

9) Calculator Misuse

• Typing 27 ^ 1 ÷ 3 without brackets gives \(9\), not \(3\). Use 27 ^ (1 ÷ 3).
• Forgetting to square roots separately in fractions.

10) Ignoring Simplification

Mistake: Leaving answers like \(\tfrac{\sqrt{50}}{10}\). Correct: Simplify: \(\sqrt{50} = 5\sqrt{2}\), so \(\tfrac{5\sqrt{2}}{10} = \tfrac{\sqrt{2}}{2}\).

By being aware of these common pitfalls, you can avoid unnecessary errors and pick up every available mark in powers and roots questions.

Worked Examples

Let’s now apply all the rules we’ve covered. Each worked example is written step by step, showing the reasoning clearly. In GCSE exams, you gain marks for method as well as the final answer, so it’s important to write full workings.

1) Basic Powers

Example A: Evaluate \(3^4\).
Solution: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Example B: Evaluate \(2^{-3}\).
Solution: Negative index means reciprocal. \(2^{-3} = \tfrac{1}{2^3} = \tfrac{1}{8}\).

2) Fractional Indices

Example C: Evaluate \(64^{1/2}\).
Solution: Fractional index = root. \(64^{1/2} = \sqrt{64} = 8\).

Example D: Evaluate \(27^{2/3}\).
Solution: Cube root first: \(\sqrt[3]{27} = 3\). Then square: \(3^2 = 9\). So \(27^{2/3} = 9\).

3) Simplifying Surds

Example E: Simplify \(\sqrt{72}\).
Solution: Factorise 72 = 36 × 2. \(\sqrt{72} = \sqrt{36}\sqrt{2} = 6\sqrt{2}\).

Example F: Simplify \(5\sqrt{50}\).
Solution: \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\). Multiply: \(5 \times 5\sqrt{2} = 25\sqrt{2}\).

4) Adding & Subtracting Surds

Example G: Simplify \(3\sqrt{3} + 7\sqrt{3}\).
Solution: Same surds, add coefficients. = \(10\sqrt{3}\).

Example H: Simplify \(4\sqrt{5} - \sqrt{5}\).
Solution: = \(3\sqrt{5}\).

5) Multiplying Surds

Example I: Simplify \(\sqrt{3} \times \sqrt{12}\).
Solution: = \(\sqrt{36} = 6\).

Example J: Expand \((\sqrt{2} + 1)(\sqrt{2} - 1)\).
Solution: Difference of squares. = \((\sqrt{2})^2 - 1^2 = 2 - 1 = 1\).

6) Rationalising Denominators

Example K: Simplify \(\tfrac{1}{\sqrt{8}}\).
Solution: Multiply by \(\tfrac{\sqrt{8}}{\sqrt{8}}\). = \(\tfrac{\sqrt{8}}{8} = \tfrac{2\sqrt{2}}{8} = \tfrac{\sqrt{2}}{4}\).

Example L: Simplify \(\tfrac{3}{2 + \sqrt{3}}\).
Solution: Multiply top and bottom by \(2 - \sqrt{3}\). Denominator: \((2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1\). Numerator: \(3(2 - \sqrt{3}) = 6 - 3\sqrt{3}\). Final: \(6 - 3\sqrt{3}\).

7) Standard Form Links

Example M: Write 0.00045 in standard form.
Solution: Move decimal 4 places right. = \(4.5 \times 10^{-4}\).

Example N: Multiply \((2 \times 10^6)(3 \times 10^{-3})\).
Solution: Multiply numbers: \(2 \times 3 = 6\). Add powers: \(10^{6 + (-3)} = 10^3\). Final: \(6 \times 10^3\).

8) Mixed Roots & Powers

Example O: Evaluate \(\sqrt[3]{16} \times 2^{2}\).
Solution: \(\sqrt[3]{16} = \sqrt[3]{8 \times 2} = 2\sqrt[3]{2}\). Multiply: \(2\sqrt[3]{2} \times 4 = 8\sqrt[3]{2}\).

Example P: Simplify \(x^{3/2} \times x^{1/2}\).
Solution: Add powers: \(x^{(3/2 + 1/2)} = x^2\).

These worked examples cover the main techniques you will need: basic indices, fractional powers, surds, rationalising, and links to standard form. Next, you can try practice questions to test your understanding.

Practice Questions — Foundation Level

These questions focus on the essential skills with powers and roots. Try them without a calculator where possible, and always check your answers by estimation or reworking in reverse.

A) Evaluating Powers

1. Work out: \(2^5\)
2. Work out: \(10^{-2}\)
3. Evaluate: \((-3)^2\)
4. Evaluate: \((-2)^3\)
5. Simplify: \(5^0\)

B) Square & Cube Roots

6. Find: \(\sqrt{121}\)
7. Find: \(\sqrt{225}\)
8. Work out: \(\sqrt[3]{27}\)
9. Work out: \(\sqrt[3]{125}\)
10. Estimate: \(\sqrt{50}\)

C) Simplifying Surds

11. Simplify: \(\sqrt{20}\)
12. Simplify: \(\sqrt{45}\)
13. Simplify: \(3\sqrt{12}\)
14. Simplify: \(2\sqrt{18}\)
15. Simplify: \(\sqrt{8} + \sqrt{32}\)

D) Adding & Subtracting Surds

16. Simplify: \(2\sqrt{3} + 7\sqrt{3}\)
17. Simplify: \(5\sqrt{2} - 3\sqrt{2}\)
18. Simplify: \(4\sqrt{7} + 2\sqrt{5}\) (leave in simplest form)
19. Simplify: \(6\sqrt{6} - 2\sqrt{24}\)

E) Multiplying & Rationalising

20. Work out: \(\sqrt{2} \times \sqrt{18}\)
21. Simplify: \(2\sqrt{5} \times 3\sqrt{5}\)
22. Rationalise: \(\tfrac{1}{\sqrt{3}}\)
23. Rationalise: \(\tfrac{4}{\sqrt{8}}\)
24. Rationalise: \(\tfrac{2}{\sqrt{5}}\)

F) Standard Form Links

25. Write 45,000 in standard form.
26. Write 0.00072 in standard form.
27. Multiply: \((3 \times 10^4)(2 \times 10^2)\).
28. Divide: \(\tfrac{5.6 \times 10^6}{7 \times 10^2}\).

Practice Questions — Higher Level

These questions extend into fractional indices, more complex surds, and rationalising binomials. They are typical of Higher GCSE exam papers.

A) Fractional and Negative Indices

1. Evaluate: \(81^{1/2}\)
2. Evaluate: \(125^{2/3}\)
3. Simplify: \(16^{-3/4}\)
4. Simplify: \(\tfrac{1}{27^{-1/3}}\)
5. Write as a single power: \(x^{2/3} \times x^{4/3}\)

B) Surds — Simplify and Combine

6. Simplify: \(\sqrt{75}\)
7. Simplify: \(\sqrt{12} + 2\sqrt{27}\)
8. Simplify: \(\sqrt{18} \times \sqrt{8}\)
9. Expand and simplify: \((\sqrt{5} + 2)(\sqrt{5} - 2)\)
10. Expand and simplify: \((3 + \sqrt{2})(3 - \sqrt{2})\)

C) Rationalising More Challenging Denominators

11. Rationalise: \(\tfrac{5}{\sqrt{7}}\)
12. Rationalise: \(\tfrac{2}{1 + \sqrt{3}}\)
13. Rationalise: \(\tfrac{4}{2 - \sqrt{5}}\)
14. Rationalise and simplify: \(\tfrac{\sqrt{3}}{\sqrt{12}}\)
15. Rationalise: \(\tfrac{7}{3 + \sqrt{2}}\)

D) Mixed Powers & Roots

16. Simplify: \((27^{1/3})^2\)
17. Work out: \(32^{4/5}\)
18. Simplify fully: \(\tfrac{x^{5/2}}{x^{1/2}}\)
19. Simplify: \((y^{3/4})^2\)
20. Write as a root: \(a^{2/5}\)

E) Standard Form with Indices

21. Write 6,700,000 in standard form.
22. Write 0.000056 in standard form.
23. Multiply: \((4.5 \times 10^3)(2 \times 10^5)\).
24. Divide: \(\tfrac{3.6 \times 10^{-2}}{1.2 \times 10^{-5}}\).
25. Simplify: \((2 \times 10^4)^2\).

Challenge Questions

These questions combine multiple skills and require careful reasoning. They are designed to stretch your understanding and prepare you for the trickiest GCSE exam problems.

1. A square has an area of \(50 \, \text{cm}^2\). Express the side length as a surd, and simplify your answer.
2. Simplify fully: \((2 + \sqrt{3})^2\).
3. Simplify: \(\dfrac{3}{2 + \sqrt{5}}\).
4. Expand and simplify: \((\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})\).
5. Solve for \(x\): \(x^{3/2} = 27\).
6. Work out: \(\dfrac{\sqrt{18} + \sqrt{50}}{\sqrt{2}}\).
7. Rationalise and simplify: \(\dfrac{5}{\sqrt{3} - \sqrt{2}}\).
8. Write \(81^{-3/4}\) as a fraction in simplest form.
9. A cube has volume \(500 \, \text{cm}^3\). Find the exact length of one edge, leaving your answer as a surd.
10. Express \(x^{5/6} \times x^{1/3}\) as a single power of \(x\).

Mini Applications

Powers and roots are not just abstract concepts — they appear in real-life contexts across mathematics, science, and computing. Here are some key applications where GCSE students will encounter them.

1) Geometry: Area and Volume Scaling

• If the sides of a square are doubled, the new area is \(2^2 = 4\) times as large. Example: A square of side 5 cm has area 25 cm². Doubling side to 10 cm gives area 100 cm².
• If the radius of a sphere is tripled, the new volume is \(3^3 = 27\) times as large. Example: A sphere of radius 2 cm has volume \(\tfrac{4}{3}\pi(2^3) = \tfrac{32}{3}\pi\). A sphere of radius 6 cm has volume \(\tfrac{4}{3}\pi(6^3) = \tfrac{864}{3}\pi\). Ratio: 27 times bigger.

2) Physics: Motion and Energy

• Speed of light is written as \(3 \times 10^8 \, \text{m/s}\) using standard form.
• Kinetic energy formula: \(E = \tfrac{1}{2}mv^2\) involves a square of velocity.
• Gravitational potential energy: proportional to \(h\), but if height is squared in equations of motion, powers become crucial.

3) Biology & Chemistry: Standard Form and Avogadro’s Constant

• Size of a bacterium: \(2 \times 10^{-6} \, \text{m}\).
• Avogadro’s constant: \(6.02 \times 10^{23}\), showing powers of 10 at work.
• pH scale: logarithmic, based on powers of 10.

4) Computing: Algorithm Complexity

• Binary numbers use powers of 2: \(2^8 = 256\), \(2^{10} = 1024\).
• Algorithm efficiency is expressed in powers: e.g. \(O(n^2)\) means the time grows with the square of the input size.

5) Finance: Compound Interest

Compound interest uses powers to model repeated percentage growth: \[ A = P\left(1 + \frac{r}{100}\right)^n \] where \(P\) is the principal, \(r\) the rate, and \(n\) the number of periods.

• £100 invested at 5% for 3 years: \(100(1.05)^3 = £115.76\).

6) Everyday Uses

• Cooking: recipe scaling (double ingredients → area doubles, volume cubes).
• Architecture: building models at half scale → volume reduced to \((1/2)^3 = 1/8\).
• Music: frequency ratios use powers of 2 (octaves are doubling frequencies).

Quick Revision Sheet

This section collects the key facts, rules, and values about powers and roots into a compact reference. Use it as a last-minute checklist before exams.

1) Index Laws

• \(a^m \times a^n = a^{m+n}\)
• \(\tfrac{a^m}{a^n} = a^{m-n}\)
• \((a^m)^n = a^{mn}\)
• \((ab)^n = a^n b^n\)
• \(\left(\tfrac{a}{b}\right)^n = \tfrac{a^n}{b^n}\)

2) Special Powers

• \(a^0 = 1\) (for \(a \neq 0\))
• \(a^{-n} = \tfrac{1}{a^n}\)
• \(a^{1/n} = \sqrt[n]{a}\)
• \(a^{m/n} = \sqrt[n]{a^m}\)

3) Surd Rules

• \(\sqrt{ab} = \sqrt{a}\sqrt{b}\)
• \(\tfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\tfrac{a}{b}}\)
• Add/subtract only like surds: \(2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}\)
• Multiply surds: \(\sqrt{a}\sqrt{b} = \sqrt{ab}\)

4) Common Squares (1–15)

\(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225\)

5) Common Cubes (1–10)

\(1, 8, 27, 64, 125, 216, 343, 512, 729, 1000\)

6) Standard Form

• Form: \(a \times 10^n\), where \(1 \leq a < 10\).
• Positive powers: large numbers (\(3.4 \times 10^6 = 3,400,000\)).
• Negative powers: small numbers (\(5.2 \times 10^{-4} = 0.00052\)).

7) Rationalising Denominators

• \(\tfrac{1}{\sqrt{a}} = \tfrac{\sqrt{a}}{a}\)
• \(\tfrac{1}{a+\sqrt{b}} = \tfrac{a-\sqrt{b}}{a^2-b}\)

8) Key Exam Tips

• Always simplify surds fully.
• Use brackets with negative numbers: \((-3)^2 = 9\).
• Don’t round until the very end — keep exact values where possible.
• Check answers by estimation (is \(\sqrt{50}\) just above 7?).

Answers — Practice Questions

Here are the answers to both the Foundation and Higher level practice sets. Check your work carefully, and if your answer is different, retrace your steps to see where the mistake happened.

A) Foundation — Evaluating Powers

1. \(2^5 = 32\)
2. \(10^{-2} = 0.01\)
3. \((-3)^2 = 9\)
4. \((-2)^3 = -8\)
5. \(5^0 = 1\)

B) Foundation — Roots

6. \(\sqrt{121} = 11\)
7. \(\sqrt{225} = 15\)
8. \(\sqrt[3]{27} = 3\)
9. \(\sqrt[3]{125} = 5\)
10. \(\sqrt{50} \approx 7.07\)

C) Foundation — Simplifying Surds

11. \(\sqrt{20} = 2\sqrt{5}\)
12. \(\sqrt{45} = 3\sqrt{5}\)
13. \(3\sqrt{12} = 6\sqrt{3}\)
14. \(2\sqrt{18} = 6\sqrt{2}\)
15. \(\sqrt{8} + \sqrt{32} = 2\sqrt{2} + 4\sqrt{2} = 6\sqrt{2}\)

D) Foundation — Adding & Subtracting Surds

16. \(2\sqrt{3} + 7\sqrt{3} = 9\sqrt{3}\)
17. \(5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}\)
18. \(4\sqrt{7} + 2\sqrt{5}\) (cannot be combined further)
19. \(6\sqrt{6} - 2\sqrt{24} = 6\sqrt{6} - 4\sqrt{6} = 2\sqrt{6}\)

E) Foundation — Multiplying & Rationalising

20. \(\sqrt{2} \times \sqrt{18} = \sqrt{36} = 6\)
21. \(2\sqrt{5} \times 3\sqrt{5} = 6 \times 5 = 30\)
22. \(\tfrac{1}{\sqrt{3}} = \tfrac{\sqrt{3}}{3}\)
23. \(\tfrac{4}{\sqrt{8}} = \tfrac{4}{2\sqrt{2}} = \tfrac{2}{\sqrt{2}} = \sqrt{2}\)
24. \(\tfrac{2}{\sqrt{5}} = \tfrac{2\sqrt{5}}{5}\)

F) Foundation — Standard Form

25. \(45,000 = 4.5 \times 10^4\)
26. \(0.00072 = 7.2 \times 10^{-4}\)
27. \((3 \times 10^4)(2 \times 10^2) = 6 \times 10^6\)
28. \(\tfrac{5.6 \times 10^6}{7 \times 10^2} = 0.8 \times 10^4 = 8 \times 10^3\)

Higher — Fractional & Negative Indices

1. \(81^{1/2} = 9\)
2. \(125^{2/3} = (5)^2 = 25\)
3. \(16^{-3/4} = \tfrac{1}{16^{3/4}} = \tfrac{1}{(16^{1/4})^3} = \tfrac{1}{2^3} = \tfrac{1}{8}\)
4. \(\tfrac{1}{27^{-1/3}} = \tfrac{1}{1/3} = 3\)
5. \(x^{2/3} \times x^{4/3} = x^{6/3} = x^2\)

Higher — Surds

6. \(\sqrt{75} = 5\sqrt{3}\)
7. \(\sqrt{12} + 2\sqrt{27} = 2\sqrt{3} + 6\sqrt{3} = 8\sqrt{3}\)
8. \(\sqrt{18} \times \sqrt{8} = \sqrt{144} = 12\)
9. \((\sqrt{5} + 2)(\sqrt{5} - 2) = 5 - 4 = 1\)
10. \((3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7\)

Higher — Rationalising

11. \(\tfrac{5}{\sqrt{7}} = \tfrac{5\sqrt{7}}{7}\)
12. \(\tfrac{2}{1 + \sqrt{3}} \times \tfrac{1 - \sqrt{3}}{1 - \sqrt{3}} = \tfrac{2(1 - \sqrt{3})}{1 - 3} = \tfrac{2 - 2\sqrt{3}}{-2} = \sqrt{3} - 1\)
13. \(\tfrac{4}{2 - \sqrt{5}} \times \tfrac{2 + \sqrt{5}}{2 + \sqrt{5}} = \tfrac{4(2 + \sqrt{5})}{4 - 5} = \tfrac{8 + 4\sqrt{5}}{-1} = -8 - 4\sqrt{5}\)
14. \(\tfrac{\sqrt{3}}{\sqrt{12}} = \tfrac{\sqrt{3}}{2\sqrt{3}} = \tfrac{1}{2}\)
15. \(\tfrac{7}{3 + \sqrt{2}} \times \tfrac{3 - \sqrt{2}}{3 - \sqrt{2}} = \tfrac{7(3 - \sqrt{2})}{9 - 2} = \tfrac{21 - 7\sqrt{2}}{7} = 3 - \sqrt{2}\)

Higher — Mixed Powers & Roots

16. \((27^{1/3})^2 = 3^2 = 9\)
17. \(32^{4/5} = (2^5)^{4/5} = 2^4 = 16\)
18. \(\tfrac{x^{5/2}}{x^{1/2}} = x^{(5/2 - 1/2)} = x^2\)
19. \((y^{3/4})^2 = y^{6/4} = y^{3/2}\)
20. \(a^{2/5} = \sqrt[5]{a^2}\)

Higher — Standard Form

21. \(6,700,000 = 6.7 \times 10^6\)
22. \(0.000056 = 5.6 \times 10^{-5}\)
23. \((4.5 \times 10^3)(2 \times 10^5) = 9.0 \times 10^8\)
24. \(\tfrac{3.6 \times 10^{-2}}{1.2 \times 10^{-5}} = 3 \times 10^3\)
25. \((2 \times 10^4)^2 = 4 \times 10^8\)

Conclusion & Next Steps

Powers and roots are among the most fundamental ideas in mathematics. They extend the simple operation of multiplication into a whole new world: squaring, cubing, fractional exponents, negative exponents, and surds. By mastering them, you gain tools that are used everywhere — in algebra, geometry, physics, finance, and computer science.

In this tutorial we have:

• Defined powers, roots, and their notation.
• Learned and practised the index laws.
• Explored negative and fractional powers, connecting them to reciprocals and roots.
• Simplified surds and learned how to add, subtract, and multiply them.
• Rationalised denominators, both simple and binomial.
• Connected powers and roots to real-world applications in science, finance, and computing.
• Practised with exam-style questions, from basic to challenge level.

Remember, the skills of simplifying, applying the laws consistently, and spotting patterns are more valuable than memorising isolated facts. The more you practise, the more natural these processes will become, and you’ll develop a strong foundation for advanced topics such as logarithms, exponential functions, and calculus.

Next Steps in Your Learning

• Review your answers regularly and revisit any weak areas.
• Link powers and roots with other topics: equations, sequences, and graphs.
• Challenge yourself with real-world problems (finance, growth/decay, algorithm efficiency).
• Practise using powers in calculator and non-calculator exam papers.

By the end of this unit, you should feel confident with powers and roots — whether it’s simplifying a surd, writing a number in standard form, or working with indices in algebra. These skills are stepping stones to higher mathematics and essential for success at GCSE and beyond.