Powers, Roots, Standard Form, and Surds

Index laws

Law	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5² = 5⁴
(aᵐ)ⁿ = aᵐⁿ	(3²)³ = 3⁶
a⁰ = 1	7⁰ = 1
a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8
a^(1/n) = ⁿ√a	27^(1/3) = 3
a^(m/n) = (ⁿ√a)ᵐ	8^(2/3) = 4

Standard form

A number in standard form is A × 10ⁿ with 1 ≤ A < 10 and n an integer.

• 45 000 = 4.5 × 10⁴
• 0.000 36 = 3.6 × 10⁻⁴

Multiplying: (3 × 10⁴) × (5 × 10³) = 15 × 10⁷ = 1.5 × 10⁸ (adjust so 1 ≤ A < 10).

Dividing: (8 × 10⁵) ÷ (2 × 10²) = 4 × 10³.

Surds (Extended)

A surd is an irrational root such as √2.

• √(ab) = √a × √b, so √50 = √25 × √2 = 5√2.
• √a × √a = a.
• Add only like surds: 3√2 + 5√2 = 8√2.

Rationalising the denominator:

• 6/√3 = 6√3/3 = 2√3
• 1/(3 − √2) = (3 + √2)/((3 − √2)(3 + √2)) = (3 + √2)/7

Quick summary

• Add powers when multiplying, subtract when dividing, multiply for a power of a power.
• Negative index: reciprocal. Fractional index: root.
• Standard form: 1 ≤ A < 10.
• Simplify surds with square factors; rationalise with the conjugate.