Index Laws 1: Basics (ks3) understand, spot, simplify

In the expression 3⁴,

what do the numbers represent?

How do we read this and what does it mean in full?

The base is 3.

It tells us which number is being multiplied.

The index (or exponent) is 4.

It tells us how many times to multiply the base.

So: 3⁴ = 3 × 3 × 3 × 3 = 81.

Read aloud: “Three to the power of four”.

What are the index laws, and when can you use them?

Can you apply them if the bases are different?

Index laws only work when the base is the same:

Multiply: 2³ × 2² = 2⁵.

Divide: 5⁶ ÷ 5² = 5⁴.

Power of a power: (3²)³ = 3⁶. Zero index: 7⁰ = 1.

Negative index: 2⁻³ = 1 ÷ 2³ = 1⁄8.

Fractional index: 16¹ᐟ² = √16 = 4.

You cannot use these laws if the bases are different (e.g., 2³ × 3²).

If the bases are different, is it ever possible to rewrite them so you can use index laws?

Try this: Simplify 4³ × 2² by changing the base.

Yes! If one base is a power of the other.

4 = 2².

So: 4³ = (2²)³ = 2⁶

Now: 2⁶ × 2² = 2⁸. The trick is to spot a hidden power — then rewrite the base so both match.

What do you do if the bases are different and can’t be rewritten?

Try simplifying: 5² × 3³

You can’t use index laws — just calculate each part: 5² = 25, 3³ = 27, Multiply: 25 × 27 = 675.

There’s no shortcut — use regular arithmetic if bases don’t match and can't be rewritten.

Which of the following can be rewritten to match bases so you can use index laws?

A. 8² × 2³

B. 5³ × 3²

C. 9² ÷ 3

D. 4² × 2³

E. 7² × 14¹

Answers: A, C, D.

A: 8 = 2³ → 8² = (2³)² = 2⁶ → 2⁶ × 2³ = 2⁹.

C: 9 = 3² → 9² = (3²)² = 3⁴ → 3⁴ ÷ 3 = 3³.

D: 4 = 2² → 4² = (2²)² = 2⁴ → 2⁴ × 2³ = 2⁷.

B and E: Bases unrelated — no rule applies.

For each expression, decide if you can apply index laws.

If so, simplify.

If not, say why.

1. 2³ × 2²

2. 3² × 5²

3. (4²)³

4. 9² ÷ 3

5. 2⁵ ÷ 8

1. Yes: 2³ × 2² = 2⁵.

2. No: different bases — can’t simplify.

3. Yes: (4²)³ = 4⁶.

4. Yes: 9 = 3² → 9² = 3⁴ → 3⁴ ÷ 3 = 3³.

5. Yes: 8 = 2³ → 2⁵ ÷ 2³ = 2².