Exponents and roots: calculation rules

Basic exponent rules

aⁿ × aᵐ = aⁿ⁺ᵐ
aⁿ / aᵐ = aⁿ⁻ᵐ
(aⁿ)ᵐ = aⁿˣᵐ
(a × b)ⁿ = aⁿ × bⁿ

Special cases

Roots

√(a × b) = √a × √b
√(a / b) = √a / √b
√a × √a = a

Worked examples

2³ × 2² = 2⁵ = 32
(3²)³ = 3⁶ = 729
5⁻² = 1/25 = 0.04
√144 = 12         (since 12 × 12 = 144)
27^(1/3) = 3      (cube root of 27)

Scientific notation

a × 10ⁿ,   with 1 ≤ a < 10
6,500,000 = 6.5 × 10⁶
0.00042 = 4.2 × 10⁻⁴

Scientific notation builds directly on powers of 10: a positive exponent shifts the decimal point right (a large number), a negative one shifts it left (a small number). Widely used in science to handle very large or very small numbers without writing out every digit.

Rationalising a denominator

1/√2 = (1 × √2) / (√2 × √2) = √2/2

Multiplying numerator and denominator by the same root removes the root from the denominator — a writing convention that makes comparing and manually computing fractions with roots easier.