Indices and Standard Form

Laws of indices

aᵐ × aⁿ = aᵐ⁺ⁿ          (multiply → add powers)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ          (divide → subtract powers)
(aᵐ)ⁿ  = aᵐⁿ            (power of a power → multiply)
a⁰ = 1
a⁻ⁿ = 1/aⁿ             (negative → reciprocal)
a^(1/n) = ⁿ√a           (fractional → root)
a^(m/n) = (ⁿ√a)ᵐ

Examples:

3⁴ × 3² = 3⁶ = 729
5⁻² = 1/25 = 0.04
16^(1/2) = √16 = 4
27^(2/3) = (³√27)² = 3² = 9

Standard form

Write numbers as A × 10ⁿ where 1 ≤ A < 10.

4 500 000   = 4.5 × 10⁶
0.00032     = 3.2 × 10⁻⁴

Big numbers → positive power; small numbers → negative power.

Calculating in standard form

• Multiply: multiply the numbers, add the powers.

(3×10⁴)(2×10³) = 6×10⁷

• Divide: divide the numbers, subtract the powers.

(8×10⁵)÷(2×10²) = 4×10³

• Add/subtract: convert to the same power first.

Exam tip

Check A is between 1 and 10. 34 × 10⁵ is not standard form — rewrite as 3.4 × 10⁶.