Exponent calculator
Enter a base and an exponent, get the power — or solve for either one.
How do I calculate an exponent?
Multiply the base by itself as many times as the exponent says: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, and 10³ = 1,000. The calculator also handles negative and fractional exponents.
What does a negative exponent mean?
One over the positive power: 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125. A negative exponent makes the number smaller, not negative.
What is a number to the power of 0?
Always 1, for any base except 0: 5⁰ = 1 and 1,000,000⁰ = 1. It keeps the rule bᵐ ÷ bⁿ = bᵐ⁻ⁿ working, since 5² ÷ 5² = 5⁰ = 1.
What is a fractional exponent?
A root. b^(1/n) is the n-th root, so 8^(1/3) = 2 and 25^(1/2) = 5; b^(m/n) is the root raised to a power: 16^(3/4) = (⁴√16)³ = 2³ = 8.
Is −2² equal to 4 or −4?
−4. The power is done before the minus sign, so −2² = −(2²) = −4, while (−2)² = 4. Odd powers keep the sign: (−2)³ = −8.
What are the laws of exponents?
For the same base: bᵐ × bⁿ = bᵐ⁺ⁿ, bᵐ ÷ bⁿ = bᵐ⁻ⁿ, and (bᵐ)ⁿ = bᵐⁿ. For a product, (ab)ⁿ = aⁿbⁿ. They hold for negative and fractional exponents as well, which is why b⁰ = 1 and b⁻ⁿ = 1 ÷ bⁿ.
Why can a negative base to a fractional power fail?
An even root of a negative number is not a real number: (−8)^(1/2) has no real value, so the calculator returns no result. An odd root does exist, since the cube root of −8 is −2, but a computer cannot tell that the decimal 0.3333 stands for one third, so a decimal power of a negative base gives no result. To get such a root, use the mode that finds the base from the result and the exponent (here 3).
The definitions behind the button
A positive integer exponent counts repeated multiplication: bⁿ is b multiplied by itself n times. Zero and negative exponents are defined so that the laws of exponents keep working: b⁰ = 1 and b⁻ⁿ = 1 ÷ bⁿ. A fractional exponent is a root: b^(m/n) is the n-th root of b raised to the m-th power. A real number raised to a real exponent is defined through the exponential and logarithm, bˣ = e^(x ln b), which needs b > 0 in general.
Worked example: growth, a negative power and a fractional power
2¹⁰ = 1,024. A balance growing 7% a year multiplies by 1.07 each year, so after 10 years it is 1.07¹⁰ = 1.9672 times the start, about 96.7% more. 2⁻³ = 0.125. For a fractional exponent, 27^(2/3) is the cube root of 27 squared: 3² = 9. A related question is the exponent itself: the logarithm answers "10 to what power is 1,000?" with 3. The real cube root of −8 is -2, which the “base from result” mode finds; typing −8 to the power 0.3333 gives no result.
Common mistakes
Forgetting that −2² means −(2²) = −4, while (−2)² = 4, is the most frequent. Another is adding exponents when the bases differ: 2³ × 3² is not 6⁵. A third is multiplying the base by the exponent: 2⁵ is 32, not 10. For large exponents the result can exceed what a calculator can show exactly; a very large answer is displayed in scientific notation and loses precision beyond about 15 significant digits.
Sources: OpenStax Elementary Algebra 2e and Intermediate Algebra 2e (properties of exponents, negative and rational exponents); OpenStax Precalculus 2e (exponential and logarithmic functions). Related: log calculator, square root calculator, scientific notation calculator and exponential growth calculator.