Fractional indices and surds

Roots as powers, and simplifying surds.

Indices & Logarithms · O/L Maths

Level 1: Square roots as powers

Know that a^1/2 = √a

Square roots undo squaring: a^1/2 = √a.

Why? a^1/2 × a^1/2 = a^1/2 + 1/2 = a¹ = a, so a^1/2 is the number that multiplies by itself to give a.

• 49^1/2
• The index 1/2 means the square root.
• 49^1/2 = √49 = 7

Level 2: General fractional indices

Use a^m/n = (ⁿ√a)ᵐ

The denominator (n) tells you the root; the numerator (m) tells you the power.

a^m/n = (ⁿ√a)ᵐ

Easiest way: take the root first, then apply the power.

• 27^2/3
• Cube root first: ∛27 = 3
• 3² = 9

Level 3: Laws with fractional indices

Use the index laws with fractions

The index laws (add, subtract, multiply) work the same way with fractional indices.

a^1/2 × a^1/2 = a¹

• x^1/3 × x^2/3
• Add the indices: 1/3 + 2/3 = 1
• x^1/3 × x^2/3 = x

Level 4: Surds

Simplify surds

A surd is a root that is not a whole number (√2, √3, √5…). Simplify by pulling out a perfect-square factor.

√a × √b = √(ab) (√a)² = a

• √12
• Find a perfect-square factor: 12 = 4 × 3
• √12 = √4 × √3 = 2√3