GCSE Mathematics 02 — Algebra

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Topics include Algebraic Foundations & Substitution, Laws of Indices, Solving Linear Equations, Rearranging Formulae & Applied Algebra, Expanding & Factorising Polynomials, Solving Quadratics, Algebraic Fractions, and Surds.

Mathematics EN

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Laws of Indices

Rules for simplifying terms involving powers, including negative, zero, and fractional indices.

Key points

Multiplication: $a^m \times a^n = a^{m+n}$ . Add the powers.
Division: $a^m \div a^n = a^{m-n}$ . Subtract the powers.
Power of a Power: $(a^m)^n = a^{m \times n}$ . Multiply the powers.
Zero Index: $a^0 = 1$ (for $a e 0$ ).
Negative Index: $a^{-n} = \frac{1}{a^n}$ . It represents the reciprocal.

Worked example

Question

Simplify

\left(\frac{8x^6}{27}\right)^{-\frac{2}{3}}

Solution

1. Handle the negative power by flipping the fraction:

\left(\frac{27}{8x^6}\right)^{\frac{2}{3}}

.
2. Apply the fractional power (cube root then square).
3. Cube root:

\sqrt[3]{27} = 3

\sqrt[3]{8} = 2

\sqrt[3]{x^6} = x^2

. Fraction becomes

\frac{3}{2x^2}

.
4. Square everything:

\frac{3^2}{(2x^2)^2} = \frac{9}{4x^4}

Common pitfalls

Multiplying bases instead of indices in $(x^2)^3$ (getting $x^5$ instead of $x^6$ ).
Thinking $x^0 = 0$ .
Applying the power to the algebra but forgetting the coefficient (e.g., $(2x)^3 = 2x^3$ instead of $8x^3$ ).

Prerequisites

Square and cube numbers
Basic arithmetic with fractions

Further resources

Index Laws
Comprehensive guide to all index laws.

Maths Genie · article