Start Free
Level 10Grade 10skill: indices· 11 min read

Indices (Powers): Rules and Examples

In short: An index (power) is shorthand for repeated multiplication, so 34 means 3×3×3×3. When the base matches, you add powers to multiply, subtract to divide, and multiply powers for a power-of-a-power. Two facts trip up strong students: a0 = 1 for any non-zero base, and 32 is 9, not 6 — you never multiply the base by the power.

How indices actually work

An index (plural indices, also called a power or exponent) tells you how many times to multiply a number by itself. In 25 the base is 2 and the index is 5, so 25 = 2×2×2×2×2 = 32. The index counts factors — it is not a multiplier. This one distinction accounts for the most common wrong answers below.

The reason indices are worth learning as rules — rather than expanding everything by hand — is that they turn slow multiplication into fast addition. If two powers share the same base, you can combine them by adding or subtracting the indices instead of writing out dozens of factors. The rules only work when the bases are identical, so your first job on any problem is to make the bases match.

  1. Check the base. Rules for indices only apply when the bases are the same, so rewrite numbers as powers of a common base where you can (8 = 23, 9 = 32).
  2. Multiplying same base: add the indices. am × an = am+n.
  3. Dividing same base: subtract the indices. am ÷ an = am-n.
  4. Power of a power: multiply the indices. (am)n = am×n.
  5. A base to the power 0 is 1: a0 = 1 (for any non-zero a).
  6. A negative index means reciprocal: a-n = 1 / an.
  7. Only evaluate to a plain number at the very end — simplify the indices first, then compute.

The laws of indices, on one card Why it works

Multiply (same base)
am × an = am+n

23 × 24 = 27 = 128

Divide (same base)
am ÷ an = am-n

56 ÷ 52 = 54 = 625

Power of a power
(am)n = am×n

(32)4 = 38 = 6561

Zero index
a0 = 1 (for any non-zero a)

170 = 1 and (-9)0 = 1

Negative index
a-n = 1 / an

2-3 = 1/8

Power of a product
(ab)n = an × bn

(2×5)2 = 22 × 52 = 100

Here is the insight that makes a0 = 1 stop feeling like a random rule: it is forced by the division law. Since an ÷ an = 1 (any non-zero thing divided by itself), and the division law says an ÷ an = an-n = a0, the two must be equal — so a0 = 1. It is not a convention someone invented; it is the only value that keeps the rules consistent.

Worked examples

Simplify (25 × 22) ÷ 24 and give the value. = 8
  1. Same base 2 throughout, so work with the indices only.
  2. Multiply: 25 × 22 = 25+2 = 27.
  3. Divide: 27 ÷ 24 = 27-4 = 23.
  4. Evaluate last: 23 = 2×2×2 = 8.
Evaluate 70 - 23 + 32. = 2
  1. Handle each power on its own before adding or subtracting.
  2. 70 = 1 (any non-zero base to the power 0 is 1 — not 0).
  3. 23 = 2×2×2 = 8 (not 6 — the index counts factors).
  4. 32 = 3×3 = 9 (not 6).
  5. Combine: 1 - 8 + 9 = 2.
Simplify (m3)4 ÷ m5. = m7
  1. Power of a power: multiply the indices, (m3)4 = m3×4 = m12. Do not add them.
  2. Divide same base: m12 ÷ m5 = m12-5 = m7.
Write 8 × 4 as a single power of 2. = 25 (= 32)
  1. The trick is to rewrite both numbers over a common base of 2.
  2. 8 = 23 and 4 = 22.
  3. So 8 × 4 = 23 × 22 = 23+2 = 25.
  4. Check: 25 = 32, and indeed 8 × 4 = 32.
Evaluate (-3)2 + (-3)3. = -18
  1. The whole -3 is the base because it is inside brackets.
  2. (-3)2 = (-3)×(-3) = 9 (an even number of negatives gives a positive).
  3. (-3)3 = (-3)×(-3)×(-3) = -27 (an odd number of negatives stays negative).
  4. Add: 9 + (-27) = -18.
Simplify 52 ÷ 54, giving a fraction. = 1/25
  1. Divide same base: subtract the indices in order, 52-4 = 5-2.
  2. A negative index means reciprocal: 5-2 = 1 / 52.
  3. 52 = 25, so the answer is 1/25.

Traps that catch strong students

Multiplying the base by the power

✗ 32 = 6

The index does not multiply the base; it counts how many copies of the base are multiplied together. Reading 32 as 3×2 gives 6, which feels natural and is exactly the tempting wrong answer.

Fix: 32 = 3×3 = 9. Say it aloud as 'two 3s multiplied', and 25 as 'five 2s multiplied' = 32, never 2×5 = 10.

wrong_operation

Stopping after one rule

✗ (x2 × x3)2 = x5

You correctly added inside the bracket, x2 × x3 = x5, then stopped — forgetting the outer power of 2 that still has to be applied. Applying one rule out of two leaves the expression half-simplified.

Fix: Finish the job: (x5)2 = x5×2 = x10. Before you write the final line, re-scan the original for any rule you have not used yet.

partial_computation

Adding indices for a power of a power

✗ (23)2 = 25 = 32

Adding the indices (3+2 = 5) mixes up two different rules. Addition is for multiplying same-base powers; a power raised to a power multiplies the indices instead.

Fix: (23)2 = 23×2 = 26 = 64. Check by expanding: (23)2 = 23 × 23 = 8 × 8 = 64.

order_of_ops_lr

Losing the sign on a negative base

✗ (-2)3 = 8

An odd number of negative factors stays negative, but it is easy to treat every power of a negative as positive. Also watch brackets: -24 means -(24) = -16, while (-2)4 = 16.

Fix: Count the negatives: (-2)3 = (-2)×(-2)×(-2) = -8. Even power → positive, odd power → negative, and only the bracketed quantity is raised to the power.

sign_error

Practise indices

Try a mixed set: simplify same-base products and quotients, chase down power-of-a-power questions, and evaluate expressions mixing zero, negative and negative-base powers. Simplify the indices first and only compute the number at the end.

Before and after this skill

First, make sure of

Frequently asked questions

Why does any number to the power 0 equal 1?

Because the division rule forces it. an ÷ an is 1 (something divided by itself), and the rule also makes it an-n = a0. Both describe the same thing, so a0 must equal 1. It holds for every non-zero base: 50 = 1, 1000 = 1, even (-3)0 = 1. (00 is left undefined.)

What is the difference between 32 and 3×2?

3×2 = 6 is ordinary multiplication. 32 = 3×3 = 9 means '3 multiplied by itself twice'. The index counts factors of the base; it is not a multiplier. Mixing these up is the single most common indices error.

When do I add indices and when do I multiply them?

Add when you multiply two powers with the same base: am × an = am+n. Multiply when you raise a power to another power: (am)n = am×n. Subtract when you divide same-base powers. Picking the wrong operation between the first two is a classic trap.

What does a negative index mean?

A negative index means take the reciprocal: a-n = 1 / an. So 2-3 = 1/8 and 5-2 = 1/25. The result is a small positive fraction, not a negative number — the minus sign lives on the index, not on the value.

Is (-2)4 the same as -24?

No — brackets decide the base. (-2)4 = (-2)×(-2)×(-2)×(-2) = 16, because the whole -2 is raised to the power. But -24 means -(24) = -(16) = -16, because only the 2 is raised and the minus sign is applied afterwards.