How to use the exponent calculator
- Choose what you want to find: the result of a power, the exponent that turns one number into another, or the base.
- Enter the base b, the number being multiplied.
- Enter the exponent n. Whole numbers, negatives (-2), decimals (0.5) and fractions (2/3) all work. A fraction keeps the exact meaning, so (-8)^(1/3) correctly gives -2.
- When solving backwards, enter the result y as well.
- Very large or very small answers switch to scientific and e notation. For whole-number powers with more than 15 digits, the exact value appears in a box you can copy.
How exponents work
The base b is the number you multiply, and the exponent n says how many copies of it to multiply together. A negative exponent flips the answer into its reciprocal. A fractional exponent p/q takes the q-th root first and then raises it to the power p, so 8^(2/3) is the cube root of 8 (which is 2) squared, giving 4. For decimal exponents that aren't simple fractions, calculators use bⁿ = e^(n × ln b).
Going backwards uses logarithms. If bⁿ = y, taking the log of both sides gives n × log b = log y, so n = log y ÷ log b. To find the base, raise both sides to the power 1/n.
Worked example
The calculator opens with 2¹⁰: ten copies of 2 multiplied together.
2 × 2 = 4, × 2 = 8, × 2 = 16, × 2 = 32, × 2 = 64, × 2 = 128, × 2 = 256, × 2 = 512, × 2 = 1,024.
Now run it backwards. Choose "The exponent" with base 2 and result 1,024: n = log(1,024) ÷ log(2) = 3.0103 ÷ 0.30103 = 10. Choose "The base" with exponent 10 and result 1,024: b = 1,024^(1/10) = 2. Because 10 is even, -2 is also a solution, since (-2)¹⁰ = 1,024 too.
Exponent rules at a glance
| Rule | Example |
|---|---|
| bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| (bᵐ)ⁿ = bᵐⁿ | (3²)³ = 3⁶ = 729 |
| (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| b⁰ = 1 | 7⁰ = 1 |
| b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 = 0.125 |
Negative bases and even roots
A negative number raised to a whole-number power is fine: (-3)² = 9 and (-3)³ = -27. Fractional exponents are different. The cube root of -8 is -2, because (-2)³ = -8, so (-8)(1/3) = -2. But no real number squared gives -4, so (-4)(1/2) has no real answer. Its value is the imaginary number 2i. The calculator tells you when an exponent needs an even root of a negative number.
Type fractional exponents as fractions rather than rounded decimals. 1/3 is exact, while 0.333 is a different number, 333/1000, whose 1000th root of a negative number has no real value.
Frequently asked questions
What is 0 to the power 0?
The calculator gives 1, which is the convention in algebra, combinatorics and most programming languages. It treats 0⁰ as an empty product. In calculus, the limit form "0⁰" is called indeterminate, which is a separate question about how two quantities approach zero.
Why does a negative exponent make the number smaller?
A negative exponent means divide instead of multiply. 2³ multiplies 1 by 2 three times to get 8, while 2⁻³ divides 1 by 2 three times to get 0.125. For a base between 0 and 1 it works the other way: 0.5⁻² = 4.
How big can the answer be?
Ordinary calculators stop at about 10³⁰⁸. This one switches to logarithms past that point, so 2¹⁰⁰⁰⁰ shows as about 1.995063117 × 10³⁰¹⁰. For whole-number bases and exponents up to about 2,000 digits, it also prints the exact digits. For exact arithmetic on even larger whole numbers, use the big number calculator.
What is the difference between an exponent and a root?
They undo each other. Squaring 6 gives 36, and the square root of 36 gives 6 back. A root is a fractional exponent: √x = x^(1/2) and ∛x = x^(1/3). The root calculator also simplifies square roots such as √72 = 6√2.
Related calculators
- Root Calculator — Square roots, cube roots and any nth root, with the simplified radical form such as √72 = 6√2.
- Log Calculator — Logarithms in base 10, base e (ln), base 2 or any base, and solve log equations for x or for the base.
- Scientific Notation Calculator — Convert numbers to scientific, E and engineering notation and back, and add, subtract, multiply or divide in a × 10ⁿ form.
- Big Number Calculator — Exact arithmetic on whole numbers with hundreds or thousands of digits: add, multiply, divide, powers, factorials and GCD.
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