Indices and roots

Examined every WJEC paper. Foundation handles integer powers; Intermediate adds the laws; Higher tackles fractional and negative indices and surd manipulation.

Index laws

For positive integers a, b and any base x ≠ 0:

• x^a × x^b = x^(a+b)
• x^a ÷ x^b = x^(a−b)
• (x^a)^b = x^(a×b)
• (xy)^a = x^a × y^a
• x^0 = 1

Negative indices

x^(−n) = 1 / x^n.

• 2^(−3) = 1 / 2^3 = 1/8
• (3/4)^(−1) = 4/3 (reciprocal)

Fractional indices

• x^(1/n) = nth root of x. So 27^(1/3) = ∛27 = 3.
• x^(m/n) = (nth root of x)^m. So 8^(2/3) = (∛8)^2 = 2^2 = 4.

The order is reversible — root first or power first — but rooting first usually gives smaller numbers.

Roots and surds

• √(ab) = √a × √b
• √(a/b) = √a / √b
• a√b + c√b = (a + c)√b (collecting "like surds")

Simplifying surds — find a square factor

√50 = √(25 × 2) = 5√2. WJEC always wants surd form a√b with smallest b.

Rationalising the denominator

To rationalise k / √a: multiply top and bottom by √a.

• 6 / √2 = 6√2 / 2 = 3√2.

For binomial denominators (Higher only): k / (a + √b) → multiply by conjugate (a − √b).

WJEC exam tip

When using fractional indices, calculate the root first, then the power — keeps numbers small. Always show the explicit step (e.g. "8^(2/3) = (∛8)^2") for the M1.