Exponent Calculator

Use the exponent calculator to raise any base to any power. A whole-number exponent means repeated multiplication: 2¹⁰ = 2 × 2 × … × 2 (10 factors) = 1,024. A negative exponent means a reciprocal, 3⁻⁴ = 1/3⁴ = 1/81. A fractional exponent means a root, 8^2/3 = (∛8)² = 2² = 4.

How to use the exponent calculator

1. On the Power tab, type the base and the exponent. Both can be whole numbers, decimals or fractions such as 2/3, and the base can be e.
2. Read the answer: an exact value when one exists (a whole number, fraction or simplified radical such as 2√3), plus the decimal, scientific notation and, for big whole numbers, the number of digits and every digit.
3. Look at the picture to see the power built step by step, then open Show steps for the full working.
4. To find a missing exponent or base in b^x = y, use the Solve tab. To simplify expressions such as (x²)³ · x⁴ ÷ x⁵, use the Exponent rules tab.

What an exponent means

In bⁿ, the base b is the number being multiplied and the exponent n says how many times it appears as a factor. The whole expression is called a power, read “b to the n” or “b to the nth power”. So 5³ = 5 × 5 × 5 = 125. The second power is called the square and the third power the cube.

Exponents grow quickly. Each extra factor of 2 doubles the value, so 2¹⁰ is 1,024 and 2¹⁰⁰ already has 31 digits. The calculator keeps every digit exactly, up to results with about 100,000 digits, and switches to scientific notation beyond that.

Exponent rules

These rules work for any nonzero base and any exponents m and n. The exponent rules calculator practices them one at a time.

Negative and fractional exponents

Negative exponents

A negative exponent never makes the answer negative. It means “one over” the positive power: 10⁻³ = 1/10³ = 0.001. You can see why from the pattern 10² = 100, 10¹ = 10, 10⁰ = 1: each step down divides by 10, so the next steps are 1/10, 1/100 and 1/1,000.

Fractional exponents

In a^m/n, the denominator n is a root and the numerator m is a power. Taking the root first keeps the numbers small: 27^2/3 = (∛27)² = 3² = 9. A decimal exponent is a fraction in disguise, so 12^0.5 = 12^1/2 = √12 = 2√3 ≈ 3.4641. Our square root simplifier shows how √12 becomes 2√3.

Negative bases

A negative base works with any whole-number exponent: an even exponent gives a positive answer, (−2)⁴ = 16, and an odd one a negative answer, (−2)⁷ = −128. With a fractional exponent, look at the denominator in lowest terms. An odd root is fine, (−8)^1/3 = −2, because (−2)³ = −8. An even root of a negative number, such as (−16)^1/4, is not a real number, because any real number raised to an even power is positive or zero.

Parentheses matter: −8² means −(8²) = −64, while (−8)² = 64. In the calculator, a minus sign in the base box always means a negative base in parentheses.

Worked examples

Example 1: a negative exponent

5⁻² = 1/5² = 1/25 = 0.04. First find 5² = 25, then take the reciprocal.

Example 2: a fractional exponent

16^3/4: the 4th root of 16 is 2, because 2⁴ = 16. Then 2³ = 8, so 16^3/4 = 8.

Example 3: solving for the exponent

For 3^x = 81, notice that 81 = 3⁴, so x = 4. When the result is not a neat power, use logarithms: for 2^x = 10, take the log of both sides to get x · log 2 = log 10, so x = log 10 ÷ log 2 ≈ 3.3219. The logarithm calculator does this kind of problem directly.

Powers quick reference

Very large powers and scientific notation

Big powers are easiest to read in scientific notation. 2¹⁰⁰⁰ has 302 digits and is about 1.0715 × 10³⁰¹. To count the digits yourself, multiply the exponent by log₁₀ of the base, round down and add 1: 1000 × log₁₀ 2 ≈ 301.03, so 302 digits. The scientific notation converter helps with numbers in this form.

Common mistakes

• Multiplying the base by the exponent. 3⁴ is 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12.
• Making a negative exponent negative. 2⁻³ is 1/8, a positive number, not −8.
• Adding exponents of different bases. The product rule needs the same base: 2³ · 3² = 8 · 9 = 72, not 6⁵.
• Adding exponents in a sum. x² + x³ cannot be combined into x⁵; only products combine.
• Dropping the parentheses. −3² = −9 but (−3)² = 9.

Exponents describe compound interest, population growth, computer memory (2¹⁰ = 1,024) and the scale of science, from 10⁻⁹ m for a nanometer to 10⁹ for a billion. For growth over time, try the exponential growth calculator.

Exponents practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

Why is any number to the power 0 equal to 1?

By the quotient rule, a³ ÷ a³ = a³⁻³ = a⁰. But any nonzero number divided by itself is 1, so a⁰ must be 1. The pattern 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1 shows the same thing: each step divides by 2.

What is 0 to the power of 0?

0⁰ is called an indeterminate form in calculus, but in algebra, counting problems and most calculators it is defined as 1. This calculator shows 1 and notes the convention. 0 to a negative power is undefined because it would mean dividing by 0.

How do you calculate a fractional exponent?

Split the fraction: the denominator is the root and the numerator is the power. For 27^2/3, take the cube root of 27 (which is 3) and square it to get 9. If the root is not a whole number, the answer is irrational, so the calculator gives a simplified radical and a rounded decimal.

Can a negative number have a fractional exponent?

Only if the denominator of the exponent, in lowest terms, is odd. (−32)^1/5 = −2 because (−2)⁵ = −32. With an even denominator, as in (−4)^1/2, the result is not a real number; it is the complex number 2i.

How do you solve for an exponent?

Write both sides as powers of the same base if you can (8^x = 4 becomes 2^3x = 2², so x = 2/3). Otherwise take logarithms: x = log y ÷ log b. The Solve tab does both and shows the check.

Related calculators

• Exponent rules calculator: practice each law of exponents
• Scientific notation converter: very large and very small numbers
• Logarithm calculator: the inverse of an exponent
• Square root simplifier: simplify radicals such as √12
• Exponential growth calculator: growth and decay over time