Exponent Calculator
- Updated
- Formula shown
- Free, no sign-up
Results
2¹⁰ = 1,024
1. 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024
| n | 2ⁿ |
|---|---|
| -3 | 0.125 |
| -2 | 0.25 |
| -1 | 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
How to use the exponent calculator
- Choose what to solve for: the result aⁿ, the base a, or the exponent n.
- Enter the numbers you know. Exponents can be whole numbers, decimals or fractions; type 2/3 rather than 0.667 for an exact two-thirds power.
- Read the answer, its scientific notation and the working. For whole-number bases, a table shows the neighboring powers.
Exponent formula
a^(p/q) = (q-th root of a)ᵖ
Solve for the base: a = x^(1/n) · Solve for the exponent: n = ln x ÷ ln a
The base a is the number being multiplied and the exponent n says how many times. A negative exponent means “one over” the positive power, and a fractional exponent p/q means take the q-th root and raise it to the power p. To solve for a missing exponent, take logarithms of both sides; to solve for a missing base, take the n-th root.
Negative bases need care. A whole-number power of a negative number is fine: (−3)³ = −27. A fractional power is real only when the root is odd: (−8)^(1/3) = −2, but (−8)^(1/2) would be the square root of −8, which is not a real number. When solving for a base with an even power, there are two real answers, such as 3 and −3 for a⁴ = 81.
Worked examples
A negative exponent: 5⁻²
- A negative exponent means a reciprocal: 5⁻² = 1 ÷ 5²
- 5² = 25, so 5⁻² = 1 ÷ 25 = 0.04
A fractional exponent: 27^(2/3)
- A fractional exponent p/q means the q-th root, raised to the power p: 27^(2/3) = (∛27)²
- ∛27 = 3
- (3)² = 9
Growth and solving for n
Something that grows by 5% a year is multiplied by 1.05 each year, so after 10 years it is 1.05¹⁰ = 1.62889 times its starting size. Working backwards, if a quantity doubles each step, the number of steps needed to pass 1,000 is n = ln 1000 ÷ ln 2 = 9.96578, so 10 doublings are enough (2¹⁰ = 1,024). And if a cube has a volume of 343, its side is the base of a³ = 343: a = 7.
Laws of exponents, checked
| Rule | Example | Long way | Using the rule |
|---|---|---|---|
| Product: aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ | 128 | 128 |
| Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² | 25 | 25 |
| Power of a power: (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ | 729 | 729 |
| Zero exponent: a⁰ = 1 | 7⁰ | 1 | 1 |
| Negative exponent: a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/2³ | 0.125 | 0.125 |
| Fractional exponent: a^(p/q) = (q-th root of a)ᵖ | 8^(2/3) = (∛8)² | 4 | 4 |
Tips and common mistakes
- −3² is not (−3)². Without brackets the power applies first, so −3² = −9 while (−3)² = 9. This calculator always treats the base you enter as a whole, including its sign.
- Exponents don't distribute over addition. (2 + 3)² = 25, not 2² + 3² = 13.
- Type fractions as fractions. 0.333 is not exactly one third, so (−8)^0.333 has no real value, while (−8)^(1/3) = −2.
- Big powers overflow. Ordinary numbers stop near 10³⁰⁸. The calculator gives exact whole-number powers up to 2,000 digits, and the big number calculator goes further.
- For square, cube and n-th roots on their own, use the root calculator; for logarithms in any base, the log calculator.
Frequently asked questions
What does a negative exponent mean?
A negative exponent means the reciprocal of the positive power: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125, and 5⁻² = 1/25 = 0.04.
How do I calculate a fractional exponent?
a^(p/q) is the q-th root of a raised to the power p. So 27^(2/3) = (∛27)² = 3² = 9, and 16^(3/4) = (⁴√16)³ = 2³ = 8.
Why is any number to the power 0 equal to 1?
Because of the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any non-zero number divided by itself is 1. 0⁰ is usually taken as 1 as well, although it is left undefined in some contexts.
How do I solve for an unknown exponent?
Take logarithms: if aⁿ = x then n = ln x ÷ ln a. For 2ⁿ = 1000, n = ln 1000 ÷ ln 2 ≈ 9.966, so it takes just under 10 doublings to reach 1,000.
Can a negative number be raised to a fractional power?
Only when the fraction has an odd denominator in lowest terms. (−8)^(1/3) = −2 because −2 × −2 × −2 = −8, but (−8)^(1/2) is the square root of a negative number, which is not real.