IGCSE Maths / Working with algebra

Fractional indices revision notes

Edexcel International GCSE Maths (4MA1), chapter 4: Working with algebra, specification point 2.1a. Free lesson.

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A fractional index tells you to take a root and then a power. The usual index laws still work with fractional powers, so you can simplify expressions by adding powers when the bases match.

What a fractional index means

In a power such as aᵐ⁄ⁿ, the bottom number tells you which root to take and the top number tells you the power. Take the root first, then apply the power. This keeps the numbers smaller as you work. aᵐ⁄ⁿ = (ⁿ√a)ᵐ A power of 1/2 means a square root. A power of 1/3 means a cube root.

How to evaluate a fractional power

For 8²⁄³, the denominator 3 means cube root and the numerator 2 means square. Since 2³ = 8, the cube root of 8 is 2. Then square the result.

How to multiply fractional powers

When multiplying powers with the same base, add their powers. The powers can be fractions, because the multiplication law still works. After adding, simplify the fraction. The bases must match. For example, x¹⁄² × y⁵⁄² cannot be simplified using the multiplication law because the bases are different.

Apply an outside power to everything

A power outside brackets applies to every part inside them, including a number written in front of a variable. For (25x⁶)¹⁄², take the square root of both 25 and x⁶. This gives 5x³, not 25x³. The number in front is inside the brackets, so it must also be affected by the outside power.

Simplify a product with different fractions

For x³⁄⁴ × x¹⁄⁴, the bases match, so add the powers. The fractions add to 1, leaving x¹, which is simply x. A whole-number power can result even when the original powers are fractions.