An exponent says how many times to multiply a number by itself. In 2⁵ the base is 2 and the exponent is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The exponent rules are shortcuts for working with powers without writing out every multiplication, and every one of them follows from that single idea. Below are all seven rules with worked examples, the mistakes that cost students the most marks, and a quick self-test. Check any answer with our free exponent calculator.
When you multiply powers with the same base, add the exponents. Writing it out shows why: 2³ × 2⁴ is (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s multiplied together, so 2⁷.
When you divide powers with the same base, subtract the exponents, because the shared factors cancel. 5⁶ ÷ 5² leaves four 5s: 5⁴ = 625.
x⁵ × x³ = x⁸
10⁷ ÷ 10⁴ = 10³ = 1,000
a² × a × a⁴ = a⁷ (a on its own is a¹)
These rules only work for the same base. 2³ × 3⁴ cannot be combined into one power.
A power raised to another power multiplies the exponents: (3²)³ means 3² × 3² × 3², which is 3⁶. Likewise, a product raised to a power gives each factor that power: (2x)³ = 2³x³ = 8x³.
The same works for fractions: (2/3)² = 2²/3² = 4/9. It does not work for sums, which is the most common exponent mistake of all (see below).
The quotient rule explains both. 5³ ÷ 5³ = 1, but subtracting exponents gives 5⁰. So any non-zero number to the power of 0 is 1.
Similarly, 2² ÷ 2⁵ = 4/32 = 1/8, and subtracting exponents gives 2⁻³. So a negative exponent means one over the positive power: 2⁻³ = 1/2³ = 1/8. It never makes the answer negative.
Reading down the table, each step divides by 2. That pattern is the simplest way to remember why 2⁰ is 1 and not 0.
| Power | Value |
|---|---|
| 2³ | 8 |
| 2² | 4 |
| 2¹ | 2 |
| 2⁰ | 1 |
| 2⁻¹ | 1/2 |
| 2⁻² | 1/4 |
| 2⁻³ | 1/8 |
An exponent of ½ means a square root, and ⅓ means a cube root: 9^(1/2) = √9 = 3 and 27^(1/3) = 3. A fraction like 2/3 combines a root and a power, and it is easiest to take the root first: 8^(2/3) = (∛8)² = 2² = 4.
For more on roots, see our square root calculator and the square numbers table.
The minus-sign trap comes from the order of operations: powers come before multiplication, and −3² means −1 × 3². Our guide to the order of operations covers it in full.
| Mistake | Correct | Why |
|---|---|---|
| 2³ × 2⁴ = 4⁷ | 2³ × 2⁴ = 2⁷ | Keep the base; only the exponents add. |
| (a + b)² = a² + b² | (a + b)² = a² + 2ab + b² | Exponents do not spread over addition. Try a = 1, b = 2: 9, not 5. |
| −3² = 9 | −3² = −9, but (−3)² = 9 | The exponent applies before the minus sign unless brackets say otherwise. |
| 5⁰ = 0 | 5⁰ = 1 | Dividing a power by itself gives 1. |
| 4⁻² = −16 | 4⁻² = 1/16 | A negative exponent means a reciprocal, not a negative number. |
| (x³)² = x⁵ | (x³)² = x⁶ | A power of a power multiplies the exponents. |
Build speed with powers in your head using Square Roots and Speed Math, and work through multi-step expressions in Order of Operations.
| Question | Answer |
|---|---|
| 3² × 3³ | 3⁵ = 243 |
| 10⁸ ÷ 10⁵ | 10³ = 1,000 |
| (2²)⁴ | 2⁸ = 256 |
| 9⁰ | 1 |
| 10⁻² | 1/100 = 0.01 |
| 16^(1/2) | 4 |
| 27^(2/3) | 9 |
| (−2)⁴ | 16 |
Add exponents when multiplying the same base, subtract when dividing, multiply when raising a power to a power, any non-zero number to the power 0 is 1, a negative exponent means a reciprocal, and a fractional exponent means a root.
Because aⁿ ÷ aⁿ = 1, and the quotient rule says it also equals a⁰. The only exception is 0⁰, which is usually left undefined or defined as 1 depending on the context.
One divided by the positive power. For example 2⁻³ = 1/2³ = 1/8. A negative exponent does not make the number negative.
Add them when multiplying two powers with the same base, such as x² × x³ = x⁵. Multiply them when raising a power to another power, such as (x²)³ = x⁶.
A square root. For example 25^(1/2) = √25 = 5.
Every exponent rule comes from one idea: an exponent counts repeated multiplication. Remember to add when multiplying, subtract when dividing and multiply for a power of a power, watch the minus-sign trap, and check your work with the exponent calculator.