GCSE Maths Practice: powers-and-roots

Understanding How to Simplify Roots Using Prime Factorisation

At Higher GCSE level, simplifying expressions involving square roots often requires using prime factorisation. This method allows you to work systematically with the powers of prime numbers, ensuring that you can simplify accurately without missing hidden factors.

Step 1 – Express Each Number in Prime Factors

Begin by breaking each number into its prime factors. Every integer can be written uniquely as a product of prime numbers. This process is called prime factorisation. It helps identify pairs of identical factors, which can be taken outside a square root.

For example, a number such as 60 can be written as \(2^2 \times 3 \times 5\). Similarly, another number can be broken down in the same way. Once both are in prime form, you can multiply them using the laws of indices.

Step 2 – Combine Factors Using the Laws of Indices

When multiplying, add the powers of identical prime bases. For instance, \(2^3 \times 2^4 = 2^{3+4} = 2^7\). This step ensures the combined number is correctly expressed in prime factor form.

In this kind of problem, you multiply the prime factor forms of two separate numbers before taking the square root. Doing so allows you to simplify without needing to multiply large numbers directly.

Step 3 – Apply the Square Root Rule

The square root of a product of powers can be simplified by halving each exponent: \(\sqrt{a^2} = a\). More generally, \(\sqrt{a^n} = a^{n/2}\). Applying this rule to every prime factor gives a much simpler result. Always check that every power is even before taking the root; if it’s not, the remaining factor stays inside the root symbol.

Worked Examples (Different Numbers)

• \(\sqrt{72} = \sqrt{2^3 \times 3^2} = 6\sqrt{2}\)
• \(\sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2}\)
• \(\sqrt{150} = \sqrt{2 \times 3 \times 5^2} = 5\sqrt{6}\)

These examples show that working with prime factors makes simplification straightforward and avoids calculator dependency.

Common Mistakes

• Multiplying the original numbers first and then struggling to simplify a large root — always simplify with factors instead.
• Forgetting to halve the powers when taking a square root.
• Leaving unmatched factors outside the root or miscounting prime powers.

Real-Life Applications

Prime factorisation and root simplification appear in many fields, including engineering, physics, and computer science. For example, simplifying square roots can make calculations for areas, diagonal lengths, and scaling problems much faster. In coding and encryption, prime factorisation underpins public-key cryptography systems such as RSA.

Quick FAQ

• Q1: Why use prime factors instead of a calculator?
  A1: Prime factors reveal the exact structure of numbers, which is essential for simplifying expressions algebraically or in surd form.
• Q2: What if a prime factor has an odd power?
  A2: The even part of the power comes out of the root, while the leftover single factor stays inside.
• Q3: Can this be extended to cube roots or higher powers?
  A3: Yes — for cube roots, divide powers by 3 instead of 2; for fourth roots, divide by 4, and so on.

Study Tip

When simplifying roots, always write out full prime factorisations before applying index rules. This makes it easier to spot even powers and identify which numbers come out of the root. Practise this with multiple pairs of numbers so the process becomes automatic under exam pressure.