Exponents and Powers Explained
7 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
An exponent tells you 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. Exponents make repeated multiplication short, and they are the foundation of scientific notation, logarithms and compound interest.
Reading and writing exponents
The small raised number is the power. “Squared” means power 2 and “cubed” means power 3. Note that 3⁴ = 3 × 3 × 3 × 3 = 81, which is not 3 × 4 = 12.
The laws of exponents
| Law | Example |
|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| (ab)ⁿ = aⁿbⁿ | (2×5)² = 4 × 25 = 100 |
| a⁰ = 1 | 9⁰ = 1 |
Negative and fractional exponents
A negative exponent means reciprocal: 5⁻³ = 1 ÷ 5³ = 1/125 = 0.008. A fractional exponent means root: 9^(1/2) = √9 = 3 and 8^(2/3) = (∛8)² = 4. Use the exponent calculator to check any of these.
Why a⁰ = 1
Follow the pattern down: 2³ = 8, 2² = 4, 2¹ = 2. Each step divides by 2, so the next value is 2 ÷ 2 = 1, which is 2⁰. The pattern continues to 2⁻¹ = 1/2, 2⁻² = 1/4.
Where exponents are used
Compound interest (see how it works), population growth, radioactive decay, computer memory (2¹⁰ = 1,024) and scientific notation such as 3 × 10⁸.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Calculate 7^4.
Show answer
7^4: 2,401
- Write 45,000,000 in scientific notation.
Show answer
Scientific notation: 4.5 × 10^7
- Find log₂(64).
Show answer
log base 2 of 64: 6
Frequently asked questions
What is 2 to the power of 10?
1,024.
Why is any number to the power 0 equal to 1?
It keeps the division rule consistent: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and a number divided by itself is 1.
What is the difference between −3² and (−3)²?
−3² = −9 (square first, then negate). (−3)² = 9.
How do I calculate a square root using exponents?
Raise to the power 1/2. 49^(1/2) = 7.
Try it yourself
Exponent CalculatorRaise any number to a power, including negative and fractional exponents.
Scientific Notation ConverterConvert numbers to scientific, engineering and standard notation.Logarithm CalculatorCalculate logs in any base with change-of-base steps.Keep learning
Written and reviewed by the SolveCalcPro editorial team. Spot a mistake? Tell us. See our editorial policy.