Exponent Calculator

How to Use This Exponent Calculator

1. Pick a tab. Power raises a base to an exponent, Solve for exponent finds x in b^x = N, Solve for base finds b, and Exponent laws compares the product, quotient and power rules.
2. Type your numbers. Whole numbers, decimals, fractions such as 1/2, and the letter e (Euler’s number) all work. Negative values are fine in the base and the exponent.
3. Click Calculate or press Enter. The Step-by-Step Solution box opens directly under the inputs.
4. Read the answer and the key values, such as the fraction form, the decimal value and scientific notation for large results.
5. Follow the numbered steps and the Check your answer line, then use Copy result or Download PDF to keep the working.
6. Click Reset to clear the inputs and start again.

Whole-number powers of whole numbers and fractions are found exactly, with no rounding. Other answers are rounded to 10 significant digits and marked with the ≈ sign.

What the Step-by-Step Solution Shows

After you press Calculate, the answer box shows the method as well as the number.

• The answer: the exact value when there is one, or a decimal marked with ≈.
• Key values: the exact and decimal forms, scientific notation for big or tiny results, and the matching log or exponent form.
• Given and formula: your inputs and the rule used, with your own numbers substituted in.
• Numbered steps: each step has a short title and a one-line reason.
• Check your answer: the reverse operation, so you can see the result is right.
• Copy result and Download PDF: Copy gives plain text for notes, and Download PDF gives a file for homework.

What Is an Exponent?

An exponent, also called a power or an index, is the small raised number that tells you how many copies of the base to multiply together. In 2⁵ the base is 2 and the exponent is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Exponents are a short way to write repeated multiplication, and they are the basis for scientific notation, compound growth and the formulas for area and volume.

Say it aloud as “2 to the power of 5” or “2 raised to the fifth power”. A power of 2 is squared and a power of 3 is cubed, because those match the area of a square and the volume of a cube.

Exponent Formula

• Exponent of 1: b¹ = b
• Exponent of 0: b⁰ = 1 for any b that is not 0
• Negative exponent: b⁻ⁿ = 1 / bⁿ
• Fractional exponent: b^p/q = (q-th root of b)^p

Laws of Exponents

These rules save a lot of work. The first three need the same base, and the last two work for any bases.

The Exponent laws tab runs the first three rules for any base and any two exponents, so you can see how adding, subtracting and multiplying exponents gives the same result as working each power out directly.

Negative Exponents

A negative exponent does not make the answer negative. It flips the base to the bottom of a fraction.

Example: 4⁻³

1. Flip to a reciprocal: 4⁻³ = 1 / 4³
2. Work out the positive power: 4³ = 4 × 4 × 4 = 64
3. Write the result: 1/64 = 0.015625

A fraction base is flipped the same way: (2/3)⁻² = (3/2)² = 9/4.

Fractional Exponents

A fractional exponent combines a root and a power. The bottom number of the fraction is the root and the top number is the power.

Example: 27^2/3

1. The denominator is 3, so take the cube root: the cube root of 27 is 3
2. The numerator is 2, so square that result: 3² = 9

A decimal exponent works too once it is written as a fraction: 0.5 is 1/2 (a square root) and 0.25 is 1/4 (a fourth root). For a plain square root, the Square Root Calculator gives the same value with its own working.

Solving for an Unknown Exponent or Base

When the unknown sits in the exponent, as in 3^x = 200, you need a logarithm: x = log₃(200), which is about 4.822. This is why exponents and logs always appear together. The Solve for exponent tab does this step for you, and the Logarithm Calculator explains the log itself.

When the unknown is the base, as in b⁴ = 81, take the matching root: b = 81^1/4 = 3. The Solve for base tab handles this for any exponent, including fractions and negatives.

Worked Examples, Step by Step

Example 1: Compound growth

An investment of 1,000 grows 5% a year. What is it worth after 10 years?

1. Each year multiplies the value by 1.05, so after 10 years the multiplier is 1.05¹⁰
2. 1.05¹⁰ is about 1.628895
3. Multiply: 1,000 × 1.628895 is about 1,628.89

Example 2: How many doublings?

A number of users doubles every month. After how many months does 1 user become 1,000,000?

1. Write the equation: 2^x = 1,000,000
2. Take the log of both sides: x = log₂(1,000,000)
3. x is about 19.93, so the 20th month is the first month above one million

Example 3: Mixing the laws

Simplify 2⁵ × 2⁻² ÷ 2¹.

1. Multiply first by adding exponents: 5 + (−2) = 3, so the expression becomes 2³ ÷ 2¹
2. Divide by subtracting exponents: 3 − 1 = 2
3. Work out the power: 2² = 4

Common Powers Table

Where Exponents Are Used

• Money: compound interest multiplies a balance by (1 + rate) once per period.
• Science: bacteria growth, radioactive decay and the spread of a signal follow powers of a fixed factor.
• Computing: memory sizes grow in powers of 2, so 2¹⁰ = 1,024 and 2²⁰ is just over a million.
• Geometry: area uses squares and volume uses cubes. See the Volume and Geometry Calculator.
• Large and small numbers: scientific notation writes numbers as a value times a power of 10. Try the Scientific Notation Calculator.

Limits and Rounding

Inputs can have up to 15 digits and a power of 10 from −100 to 100. Whole-number powers of whole numbers and fractions are worked out exactly with big-integer arithmetic. Answers beyond about 10³⁰⁰ or below 10⁻³⁰⁰ are out of range. A negative base only gives a real answer for whole-number exponents or fractions with an odd denominator. Write a fractional exponent as a fraction such as 2/3 rather than a long decimal when you want an exact root.

Frequently Asked Questions

What is an exponent?

An exponent tells you how many times to multiply a base by itself. In 4³ the base is 4 and the exponent is 3, so 4³ = 4 × 4 × 4 = 64.

How do I calculate a negative exponent?

Take the reciprocal of the base, then use the positive exponent. For example, 5⁻² = 1/5² = 1/25 = 0.04. The Power tab shows each of these steps.

What does a fractional exponent mean?

The denominator is a root and the numerator is a power. So 16^(1/2) is the square root of 16, which is 4, and 8^(2/3) is the cube root of 8 squared, which is 4.

What is anything to the power of 0?

Any non-zero number to the power 0 equals 1. For example, 7⁰ = 1. The expression 0⁰ is left undefined in most school courses.

How do I solve for an unknown exponent?

Use a logarithm. If 2^x = 64, then x = log₂(64) = 6. The Solve for exponent tab does this and shows the steps.

How do I solve for an unknown base?

Take the root that matches the exponent. If b⁴ = 81, then b = 81^(1/4) = 3. The Solve for base tab does this for any exponent, including fractions.

What are the main laws of exponents?

With the same base, multiplying adds the exponents, dividing subtracts them, and a power of a power multiplies them. The Exponent laws tab shows all three for your numbers.

Can the base be negative?

Yes. A negative base gives a real answer for whole-number exponents, where an even exponent gives a positive result and an odd one a negative result. It also works for fractional exponents with an odd denominator, such as (−8)^(1/3) = −2.

What is the difference between an exponent and a logarithm?

They are inverse operations. An exponent builds a number from a base and a power, while a logarithm recovers the power. The Logarithm Calculator page covers logs in detail.

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