The laws of indices

For powers of the same number:

aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ

Why: 5³ × 5² = (5 × 5 × 5) × (5 × 5) = 5⁵. Counting the fives is adding the indices.

And a⁰ = 1, because 5³ ÷ 5³ = 5⁰ and anything divided by itself is 1.

Deal with the numbers and the letters separately.

1. 3x⁴ × 4x⁵ = (3 × 4) × x⁴⁺⁵ = 12x⁹
2. (2x³)⁴ = 2⁴ × x³ˣ⁴ = 16x¹²
3. 12x⁷y³ ÷ 3x²y = 4x⁵y²

The classic slip: (2x³)⁴ = 8x¹². The 2 is raised to the power 4 too, so it becomes 16.