Multiplying Powers

When multiplying powers with the same base, add the indices.

Formula: aᵐ × aⁿ = aᵐ⁺ⁿ

Example: 3² × 3⁴ = 3⁶ = 729

Dividing Powers

When dividing powers with the same base, subtract the indices.

Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Example: 5⁷ ÷ 5³ = 5⁴ = 625

Power of Zero

Any number raised to the power of 0 equals 1.

Formula: a⁰ = 1

Example: 7⁰ = 1 100⁰ = 1

Power of One

Any number to the power of 1 is just the number itself.

Formula: a¹ = a

Example: 15¹ = 15

Common Mistake

Index laws ONLY work when the base (the big number) is the same.

Base 10 Powers

Powers of 10 follow a simple pattern: 10ⁿ is 1 followed by n zeros.

Example: 10³ = 1000 10⁶ = 1,000,000

Squaring and Cubing

Index 2 means square (multiply by itself once). Index 3 means cube (multiply by itself twice).

Formula: x² = x × x, x³ = x × x × x

Example: 5³ = 5 × 5 × 5 = 125

Negative Indices

A negative index means the reciprocal (1 over) the positive power.

Formula: a⁻ⁿ = 1/aⁿ

Example: 2⁻³ = 1/2³ = 1/8

Fractional Indices (Roots)

The denominator is the root, the numerator is the power.

Formula: a^(1/n) = ⁿ√a

Example: 9^(1/2) = √9 = 3 64^(1/3) = ³√64 = 4

Complex Fractional Indices

Apply the root first (bottom number), then the power (top number).

Formula: a^(m/n) = (ⁿ√a)ᵐ

Example: 27^(2/3) = (³√27)² = 3² = 9

Power of a Power

When raising a power to another power, multiply the indices.

Formula: (aᵐ)ⁿ = aᵐˣⁿ

Example: (2³)⁴ = 2¹² = 4096

Fractional Bases

Apply the power to both numerator and denominator.

Example: (2/3)⁻² = (3/2)² = 9/4

Combining Laws

In complex questions, follow BIDMAS. Apply power-to-power first, then multiply/divide.

Example: (x²)³ × x⁴ = x⁶ × x⁴ = x¹⁰

Base Conversion

Sometimes you need to change the base to use index laws.

Example: 9⁴ = (3²)⁴ = 3⁸