How do you calculate an exponent by hand?

For a whole-number exponent, multiply the base by itself that many times: 3⁴ = 3 × 3 × 3 × 3 = 81. For a negative exponent, work out the positive power and take 1 over it: 3⁻⁴ = 1 ÷ 81. For a fraction, take the root: 81^¼ = 3 because 3⁴ = 81. For anything else — 1.03²⁰ — a calculator's xʸ or ^ key, or logs: 20 × log(1.03) = 0.2567, and 10^0.2567 = 1.806.

What is 1.03 to the power of 20, or 1.05 to the power of 20?

1.03²⁰ = 1.8061 and 1.05²⁰ = 2.6533. These are compound-growth factors: at 3% a year, money grows by 80.6% over 20 years; at 5% it grows by 165%. Enter the rate as 1 + rate and the years as the exponent.

What does a negative exponent mean?

Divide by that many copies of the base instead of multiplying: 10⁻² = 1 ÷ 10² = 0.01, 2⁻³ = 1 ÷ 8 = 0.125. It is the same rule that makes the sequence 2³, 2², 2¹, 2⁰, 2⁻¹ go 8, 4, 2, 1, ½ — each step divides by 2.

How do I find the exponent if I know the base and the answer?

Take logarithms: x = log(answer) ÷ log(base). For 5ˣ = 625, log(625) ÷ log(5) = 2.79588 ÷ 0.69897 = 4. Choose 'The exponent' mode, enter the base and the result, and the page does the division.

Why is anything to the power 0 equal to 1?

Because of the quotient rule: bⁿ ÷ bⁿ = bⁿ⁻ⁿ = b⁰, and anything divided by itself is 1. The one exception is 0⁰, which the rule cannot reach because 0 ÷ 0 is undefined.

Exponent Calculator (Powers, Negative and Fractional Exponents, Solve for x)

Any base to any power with the working, or the exponent that turns a base into a result.

Inputs

Find

Base

-1,0001,000

Exponent

-100100

Result (to solve for x)

01,000,000,000,000,000

Include CalculateEveryday mark

2^10

1,024

Working: 2 multiplied by itself 10 times = 1,024
Growth factor: 2 per step, 10 steps: ×1,024 overall, 102,300% more

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What each Exponent input means

Find: Raise a base to a power, or find the exponent: the x in base^x = result.
Base: The number being multiplied by itself. Negative bases work with whole-number exponents; for growth, enter 1 plus the rate, such as 1.07 for 7%.
Exponent: The power: how many times the base is multiplied by itself. Negative and decimal exponents are allowed — 2⁻³ = 0.125 and 9^0.5 = 3.
Result (to solve for x): In 'The exponent' mode, the result you want to reach. It must be positive, and the base must be positive and not 1.

Exponent formula

bⁿ = b × b × … × b (n times) · b⁻ⁿ = 1 ÷ bⁿ · b^(1/n) = ⁿ√b · b⁰ = 1 (b ≠ 0) · solve bˣ = y: x = log(y) ÷ log(b)

b^{-n} = \frac{1}{b^{n}}, \quad b^{1/n} = \sqrt[n]{b}, \quad b^{x} = y \iff x = \frac{\log y}{\log b}

An exponent counts how many times the base is multiplied by itself: 2¹⁰ is ten 2s multiplied together, 1,024. The rules extend that idea past whole numbers. A negative exponent means divide instead of multiply — 2⁻³ is 1 ÷ 2³ = 0.125. A fractional exponent is a root: 9^½ is the square root of 9, and 8^⅓ the cube root. Anything except 0 raised to the power 0 is 1. The page shows the multiplication written out for small whole exponents, explains the reciprocal or root for the others, and switches to scientific notation when the answer is huge or tiny. Because a power like 1.03²⁰ is exactly how compound growth works — 3% a year for 20 years multiplies money by 1.806 — the growth-factor line says what the result means as a percentage. The second mode runs backwards: given the base and the result, it finds the exponent with logarithms, which is how you answer 'how many doublings to reach 1,024?'

Exponent worked example

2¹⁰: ten 2s multiplied together, 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024. Reversed, 2ˣ = 1,024 gives x = log(1,024) ÷ log(2) = 3.0103 ÷ 0.30103 = 10. A finance example: 1.03²⁰ = 1.806, so a sum growing at 3% a year is 80.6% bigger after 20 years; 1.05²⁰ = 2.653, so at 5% it more than doubles.

More Exponent examples

How many doublings reach 1,000? 2ˣ = 1,000 gives x = 9.965784, so 10 doublings are needed, and they take 1 to 1,024 — just past it.

1.07¹⁰ = 1.967151: money growing 7% a year almost doubles in 10 years, which is the rule of 72 at work (72 ÷ 7 ≈ 10.3 years).

Where the Exponent calculator can give the wrong answer

0⁰ and 0 to a negative power are undefined, and the page says so rather than showing 1 or infinity. 0 to any positive power is 0.
A negative base with a fractional exponent is refused. (−8)^½ is not a real number at all; (−8)^⅓ has a real answer, −2, but the general rule cannot tell that case from the first, so take the cube root of 8 and put the sign back yourself.
Very large results overflow double precision above about 1.8 × 10³⁰⁸ (2¹⁰²⁴ is too big); very small ones underflow to 0 below about 10⁻³²⁴. The scientific-notation line shows results outside 10⁻⁴ to 10⁶.
Decimal exponents are computed in floating point, which rounds at about the sixteenth digit; results are shown to six decimals, so 10^0.30103 reads 2 rather than 1.99999996.
Solving for the exponent needs a positive base other than 1 and a positive result; there is no real x with 2ˣ = −4, and 1ˣ is always 1.

Spotted a mistake? Tell us

Exponent FAQ

: For a whole-number exponent, multiply the base by itself that many times: 3⁴ = 3 × 3 × 3 × 3 = 81. For a negative exponent, work out the positive power and take 1 over it: 3⁻⁴ = 1 ÷ 81. For a fraction, take the root: 81^¼ = 3 because 3⁴ = 81. For anything else — 1.03²⁰ — a calculator's xʸ or ^ key, or logs: 20 × log(1.03) = 0.2567, and 10^0.2567 = 1.806.
: 1.03²⁰ = 1.8061 and 1.05²⁰ = 2.6533. These are compound-growth factors: at 3% a year, money grows by 80.6% over 20 years; at 5% it grows by 165%. Enter the rate as 1 + rate and the years as the exponent.
: Divide by that many copies of the base instead of multiplying: 10⁻² = 1 ÷ 10² = 0.01, 2⁻³ = 1 ÷ 8 = 0.125. It is the same rule that makes the sequence 2³, 2², 2¹, 2⁰, 2⁻¹ go 8, 4, 2, 1, ½ — each step divides by 2.
: Take logarithms: x = log(answer) ÷ log(base). For 5ˣ = 625, log(625) ÷ log(5) = 2.79588 ÷ 0.69897 = 4. Choose 'The exponent' mode, enter the base and the result, and the page does the division.
: Because of the quotient rule: bⁿ ÷ bⁿ = bⁿ⁻ⁿ = b⁰, and anything divided by itself is 1. The one exception is 0⁰, which the rule cannot reach because 0 ÷ 0 is undefined.