Algebraic indices

Revise Algebraic indices for Mathematics D 4024 (O Level) — revision notes, 46 practice questions and instant AI marking.

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Cambridge O Level Mathematics (Syllabus D) 4024

Algebraic indices

The big idea: The rules of indices work just as well with letters as with numbers. Deal with the numbers and the letters separately, and to solve an equation with the unknown in the power, write both sides as powers of the same base.

Summary — What This Chapter Covers

Use positive, zero, negative and fractional indices in algebra.
Simplify expressions such as 3x⁻⁴ × (2/3)x^1/2 and (2x⁵/3)³.
Solve equations such as 32^x = 2 and 5^{x + 1} = 25^x by matching bases.
Logarithms are not required.

1. Simplifying with the Index Rules

Syllabus 2.4 · Paper 1 & 2

Numbers with numbers, letters with letters

Multiply or divide the numerical coefficients as ordinary numbers; use the index rules on each letter. A power outside a bracket applies to every factor inside, including the number.

Rules x^m × xⁿ = x^m+n · x^m ÷ xⁿ = x^m−n · (x^m)ⁿ = x^mn · (ab)ⁿ = aⁿbⁿ · x⁻ⁿ = 1/xⁿ

Worked Example

Simplify (a) 3x⁻⁴ × (2/3)x^1/2 (b) (2/5)x^1/2 ÷ 2x⁻² (c) (2x⁵/3)³

Practice Question 1

Simplify (a) (16x⁸)^3/4 (b) (27x⁶)^−1/3 (c) (6x³ × x⁻¹)/(3x^1/2)

2. Solving Equations with Indices

Syllabus 2.4 · Paper 1 & 2

Same base on both sides, then equate the powers

Write both sides as a power of the same prime base (2, 3, 5, …). When the bases are equal, the indices are equal, which gives an ordinary linear equation.

Worked Example

Solve (a) 32^x = 2 (b) 5^{x + 1} = 25^x

Practice Question 2

Solve (a) 9^x = 27 (b) 4^{x − 1} = 8^x (c) 8^{x + 1} = 16^x

Memorise

Coefficients

Bracket to a power

⁵ ³ ³ ¹⁵ ¹⁵

Index equations

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Algebraic indices