Exponent Calculator
Raise any number to a power (including negative and fractional exponents) with the working shown.
Details
Negative and fractional exponents are allowed.
2¹⁰ =
1,024
This raises any number to any power, including negative and fractional exponents.
It returns the answer as a decimal and in scientific notation, which matters because powers grow large enough to become unreadable very quickly.
What an exponent is
An exponent tells you how many times to multiply a number by itself. In 2⁵, the 2 is the base and the 5 is the exponent, so it means 2 × 2 × 2 × 2 × 2 = 32.
The number gets big fast. 2¹⁰ is already 1,024, and 2²⁰ is over a million. That runaway growth is why exponents describe compound interest, population growth and computer storage.
Three cases look strange until you see the pattern behind them: a power of zero gives 1, a negative power gives a fraction, and a fractional power is a root.
The exponent counts the base, not the multiplications. 2⁵ uses five 2s, which means four multiplication signs.
What to enter
- Base (a)
- The number being multiplied. It can be negative or a decimal.
- Exponent (n)
- How many times. Whole numbers, negatives and fractions all work, and each behaves differently.
The cases that look odd
- a⁰ = 1
- Anything to the power zero is 1. Following the pattern down — 2³=8, 2²=4, 2¹=2 — each step halves, so the next is 1.
- a⁻ⁿ = 1 ÷ aⁿ
- A negative exponent flips it into a fraction. 2⁻³ = 1 ÷ 8 = 0.125. It does not make the answer negative.
- a^(1/2) = √a
- A fractional exponent is a root. The power 1/2 is a square root, 1/3 is a cube root.
- a^(m/n)
- Take the nth root, then raise to the mth power. 8^(2/3) = (∛8)² = 4.
- 0⁰
- Usually defined as 1 by convention, though it is genuinely ambiguous and some contexts leave it undefined.
What this assumes
Very large or very small results are shown in scientific notation, because the full digits become unreadable.
A negative base with a fractional exponent has no real answer, since it would mean an even root of a negative number.
How to calculate a power
For whole numbers, multiply repeatedly. The other cases follow from three short rules.
- a
- The base
- n
- How many copies of the base are multiplied
Whole exponents: multiply. 2⁵ means five 2s multiplied: 2, 4, 8, 16, 32.
Negative exponents: flip. Work out the positive power, then take one over it. 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125.
Fractional exponents: root it. The bottom of the fraction is the root, the top is the power. 8^(2/3) means cube root of 8, squared, which is 4.
Multiplying powers: add exponents. 2³ × 2⁴ = 2⁷. Dividing subtracts them, and a power of a power multiplies them.
See a worked example: why 2⁰ = 1
- The pattern
- 2³, 2², 2¹, 2⁰
2³ = 8. 2² = 4. 2¹ = 2.
Each step down divides by 2: 8 → 4 → 2.
Continuing the pattern, the next step is 2 ÷ 2 = 1.
So 2⁰ = 1, and the same reasoning works for any base. Going one further gives 2⁻¹ = 0.5, which is where negative exponents come from.
2⁰ = 1
Frequently asked questions
Follow the pattern downwards. 2³ = 8, 2² = 4, 2¹ = 2, and each step divides by 2. The next step is 2 ÷ 2 = 1.
The rule for dividing powers gives the same answer: 2³ ÷ 2³ = 2⁰, and any number divided by itself is 1.
No. It makes it a fraction. 2⁻³ = 1 ÷ 2³ = 0.125, which is positive.
The sign of the answer comes from the base, not the exponent. (−2)³ is negative because the base is negative.
It is a root. The denominator says which root: a^(1/2) is a square root and a^(1/3) is a cube root.
With both parts, 8^(2/3) means take the cube root of 8, which is 2, then square it, giving 4. Rooting first keeps the numbers smaller.
They are the same relationship read in opposite directions. An exponent asks 'what do I get?'; a logarithm asks 'what power was used?'.
2⁵ = 32, and log₂(32) = 5. Each undoes the other, exactly as multiplication and division do.
Multiplying adds the exponents: 2³ × 2⁴ = 2⁷. Dividing subtracts them: 2⁷ ÷ 2³ = 2⁴.
A power raised to a power multiplies them: (2³)⁴ = 2¹². All three only work when the bases match.
Problems people actually run into
Multiplying the base by the exponent
2⁵ is not 10. It is 32. The exponent counts how many times to multiply the base by itself, and it is not a multiplier.
The mistake is easiest to spot with a large exponent: 2¹⁰ is 1,024, nowhere near 20. It slips through most often on small numbers where the two answers are closer.
Applying a power to a sum
(a + b)² does not equal a² + b². Squaring 3 + 4 gives 49, while 9 + 16 gives 25.
Powers distribute over multiplication, so (ab)² really is a²b², but never over addition. This is one of the most persistent errors in algebra.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 4, 2026