What this calculator does
Work out any power of a number, with each step explained. An exponent, the small raised number, says how many times to multiply a number by itself. So 2¹⁰ means ten 2s multiplied together, which is 1,024. The number being multiplied (here 2) is called the base.
Exponents can also be negative, decimals or fractions. A negative exponent means 1 divided by the power (its reciprocal). A fraction like 8^(2/3) means the cube root of 8, squared. The calculator handles all of these and tells you which rule it used.
It also works backward. Say you know the base and the answer, and want the exponent. For example: 2 to what power is 1,024? Or how many 5% growth periods does it take to double? The solve mode finds the exponent using logarithms (logs), which undo powers.
Where it's used
Why this math matters outside the classroom:
- Compound growth. A $1,000 deposit earning 5% a year grows to 1000 × 1.05¹⁰, about $1,628.89, after 10 years.
- Computer memory. Programmers know that 2¹⁰ = 1,024, so a kibibyte holds 1,024 bytes and 2²⁰ bytes make a mebibyte.
- Bacteria doubling. A culture of 500 cells that doubles every hour reaches 500 × 2⁶ = 32,000 cells after 6 hours.
- Small measurements. Science students read 10⁻³ as 1 ÷ 1,000, so a 2.5 mm part is 2.5 × 10⁻³ m long.
- Scale models. Engineers find that 8^(2/3) = (∛8)² = 4, the factor a model's surface area grows by when its volume grows 8 times.
Grade level and standards: Grades 8–9; Common Core State Standards for Mathematics 8.EE.A.1, HSN.RN.A.2. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose a mode: raise to a power, use a fraction exponent p/q, or solve for the exponent.
- Enter the base. To raise to a power, enter the exponent; it can be negative or a decimal.
- For a fraction exponent, enter the top number (numerator) and bottom number (denominator) as separate whole numbers. That keeps roots such as 1/3 exact.
- To solve for the exponent, enter the base and the result you want the power to equal. Both must be positive, and the base can't be 1.
- Read the answer and the steps. If a combination has no answer, such as 0 to a negative power, the calculator tells you why.
What the inputs mean
- Exponent (x)
- A decimal such as 0.5 works only for a base that's positive or zero. For a negative base with a fraction power, switch to the fraction exponent mode. Then the calculator can tell whether the root is odd.
- Exponent denominator (q)
- The root to take. The fraction is reduced first, so 2/6 is treated as 1/3. A negative base then needs an odd denominator.
- Result (bⁿ)
- The value of bⁿ you want to reach. It must be positive, because a positive base raised to any power is positive.
Formula and method
bⁿ = b × b × … × b (n factors) b⁻ⁿ = 1 ÷ bⁿ, b⁰ = 1 b^(p/q) = (q-th root of b)^p b^x = e^(x · ln b) for non-integer x and b > 0 n = ln(y) ÷ ln(b) solves bⁿ = y
In the formulas, b is the base and n is a whole-number exponent. p/q is a fraction exponent, with p on top and q on the bottom. x is any decimal exponent, and y is the result you want in solve mode. e is a special number, about 2.718, and ln means the natural log, the log that goes with e.
Whole-number exponents are repeated multiplication. The other rules come from keeping one law true: bᵐ × bⁿ = bᵐ⁺ⁿ (multiplying powers adds their exponents). For that to work, a zero exponent must give 1, and a negative exponent must give 1 divided by the power. An exponent of 1/q must give the q-th root, because q copies of b^(1/q) multiplied together have to give back b.
For any decimal exponent, the calculator uses the rule b^x = e^(x ln b). To solve bⁿ = y, it takes the natural log of both sides. That turns the exponent into an ordinary number you multiply by: n · ln b = ln y.
Worked example
Evaluate 8^(2/3), and then find the exponent n for which 2ⁿ = 1,024.
- 8^(2/3) means the cube root of 8, squared. The cube root of 8 is 2.
- Squaring gives 2² = 2 × 2 = 4.
- For 2ⁿ = 1,024, take the natural log of both sides: n × ln 2 = ln 1,024.
- n = 6.9314718056 ÷ 0.6931471806 = 10.
Result: 8^(2/3) = 4, and 2¹⁰ = 1,024, so the missing exponent is 10.
Understanding the result
Say a positive base gives a result between 0 and 1. That means the exponent was negative, or the base itself was below 1. Each step further into the negatives divides by the base once more.
When you solve for the exponent, an answer that isn't a whole number is normal. For example, 3ⁿ = 20 needs n ≈ 2.7268, which sits between 3² = 9 and 3³ = 27. In growth problems, the exponent is often a number of periods. So round it up if you need a whole number of periods.
Assumptions and limitations
- 0⁰ is counted as 1. That's the usual agreement in algebra, in counting problems and in most programming languages. Some parts of calculus leave it undefined.
- Only real-number results are given. Negative bases work with whole-number exponents. They also work with fraction exponents whose bottom number, once the fraction is reduced, is odd.
- Results bigger than about 1.8 × 10³⁰⁸, or closer to zero than about 5 × 10⁻³²⁴, are too extreme to store. The calculator shows an error for those.
- Computers store decimals in a way that isn't always exact. So an exponent like 0.3333 is treated as exactly 3333/10000, not as 1/3. Use the fraction mode for exact roots.
- Complex results, such as (−4)^0.5 = 2i, are beyond what this tool does. (Here i is the imaginary number whose square is −1.)
Frequently asked questions
Why is any number to the power 0 equal to 1?
When you divide powers of the same base, you subtract the exponents: bⁿ ÷ bⁿ = bⁿ⁻ⁿ = b⁰. But the left side is a number divided by itself, which is 1. So b⁰ has to be 1 for every base except 0.
What does a negative exponent mean?
It means 1 divided by the positive power. So 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125. A negative exponent never makes the result negative on its own. When the base is bigger than 1, it makes the result smaller.
Why is 0 to a negative power undefined?
0⁻² would be 1 ÷ 0², which is 1 ÷ 0. Dividing by zero has no answer, so the calculator shows an error rather than infinity.
Why can't I raise a negative number to 0.5?
Raising to 0.5 is the same as taking a square root. No real number multiplied by itself gives a negative number. Fraction powers with an odd bottom number do have real answers, like (−8)^(1/3) = −2. Use the fraction exponent mode for those.
How do I find how many periods it takes to double at a given growth rate?
Solve (1 + r)ⁿ = 2 for n, where r is the growth rate as a decimal. At 5% per period, use base 1.05 and result 2. Then n = ln 2 ÷ ln 1.05 ≈ 14.21, so it takes just over 14 periods.
How this calculator is checked
Method: Laws of exponents; b^x = e^(x ln b) for real exponents; natural logarithm for solving bⁿ = y. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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