GCSE Maths Practice: powers-and-roots

Working with Fractional Indices and the Laws of Indices

Once you understand how fractional indices represent powers and roots, the next step is to combine them using the laws of indices. These laws apply universally to integers, fractions, and even negative exponents. Mastery of these rules helps simplify complex expressions quickly and is vital in algebra, surds, and exponential equations.

The Three Core Index Laws

• Multiplication Law: When multiplying powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Division Law: When dividing powers with the same base, subtract the exponents: \(a^m \div a^n = a^{m-n}\).
• Power Law: When raising one power to another, multiply the exponents: \((a^m)^n = a^{mn}\).

These rules remain valid even when the exponents are fractions. For instance, adding or subtracting fractional indices follows normal fraction arithmetic — the denominators must match before combining them.

Step-by-Step Method

1. Check that the bases are identical. You can only combine powers with the same base directly.
2. Apply the correct index law depending on the operation — add for multiplication, subtract for division.
3. Simplify the resulting fractional exponent by adding or subtracting numerators while keeping the same denominator.
4. If necessary, convert the fractional power to root form to simplify further.

Worked Examples (Different Bases)

• \(4^{3/2} \div 4^{1/2} = 4^{(3/2 - 1/2)} = 4^1 = 4\)
• \(9^{5/2} \div 9^{3/2} = 9^{(5/2 - 3/2)} = 9^1 = 9\)
• \(27^{4/3} \div 27^{1/3} = 27^{(4/3 - 1/3)} = 27^{3/3} = 27\)

Each example demonstrates that fractional indices behave exactly like integer exponents under the index laws.

Common Mistakes

• Using the wrong operation for exponents — adding instead of subtracting when dividing.
• Trying to combine powers with different bases (e.g., \(2^a \times 3^a\) cannot be simplified without evaluating).
• Forgetting to simplify fractional arithmetic properly, such as miscalculating \(4/3 - 1/3\).
• Switching between fraction and root forms inconsistently during simplification.

Real-Life Applications

The laws of indices are used in many real-world problems. In physics, they simplify equations for wave motion, sound intensity, and electrical power. In computing, algorithms with time complexity expressed as powers use these laws to combine or simplify terms. In finance, interest formulas often require manipulation of powers and roots, especially when comparing annual and monthly compounding rates. Understanding these principles allows you to simplify complex relationships before applying a calculator or further algebraic steps.

Quick FAQ

• Q1: Can I apply these rules to negative or fractional exponents?
  A1: Yes — the laws of indices work for all rational exponents.
• Q2: Why must the bases be the same?
  A2: Because the laws come from grouping identical factors — if the bases differ, the grouping principle fails.
• Q3: Do the laws apply to roots written as radicals?
  A3: Yes. For example, \(\sqrt{a}\times\sqrt{a^3}=a^{1/2+3/2}=a^2\).

Study Tip

Practise adding and subtracting fractional exponents mentally. Start with denominators of 2, 3, and 4 to build fluency. Always check that the final exponent makes sense — for division, the result should have a smaller power than the original numerator. Keep all work in exact form until the end to avoid rounding errors and to show complete understanding of index laws.