Exponent Calculator

What an exponent does

An exponent says how many times a number is used as a factor in a product. In the expression bⁿ, the number b is the base and n is the exponent (also called the power or index). The whole expression is called a power of b.

b^n = b × b × … × b (n factors of b)

For example, 3⁴ means 3 × 3 × 3 × 3, which is 9 × 9 = 81. An exponent of 2 is read "squared" and an exponent of 3 is read "cubed", because they give the area of a square and the volume of a cube with that side length.

Exponents are not limited to whole numbers. Negative, decimal and fractional exponents all have meanings that follow from a few simple rules, covered below, and this calculator accepts any real number in both boxes.

How to use the exponent calculator

1. Type the base, the number you want to raise, into the first box. It can be positive, negative or a decimal.
2. Type the exponent into the second box. It can also be positive, negative or a decimal. Use a dot as the decimal mark.
3. Press Calculate. The answer appears in the Power row.

Whole-number answers are written out in full with thousands separators, so 2¹⁰ appears as 1,024. Answers with a fractional part show at least 8 decimal places and at most 14, for example 2^-3 appears as 0.12500000 and the square root of 2 as 1.41421356237310.

Two special answers can appear. nan (not a number) means the power has no real value, which happens when a negative base meets a fractional exponent. inf means the result is larger than any number a computer can store, roughly 1.8 × 10³⁰⁸.

A base of zero is not accepted. Zero raised to a positive power is simply 0, zero raised to a negative power is undefined because it would mean dividing by zero, and 0⁰ is left undefined in most textbooks.

The rules of exponents

Every property of powers comes from the definition above. These are the ones you will use most often, each with a small example you can check in the calculator.

The zero-exponent rule is a direct consequence of the quotient rule. Dividing any non-zero number by itself gives 1, and aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, so a⁰ has to equal 1.

Worked examples

A positive whole exponent. 7³ = 7 × 7 × 7 = 49 × 7 = 343. Enter 7 as the base and 3 as the exponent.

A negative exponent. 2⁻⁵ = 1 ÷ 2⁵ = 1 ÷ 32 = 0.03125. The calculator shows 0.03125000, with trailing zeros up to 8 decimals.

A fractional exponent. 81^3/4 means the fourth root of 81, which is 3, raised to the third power: 3³ = 27. You can enter the exponent as 0.75.

Compound growth. An account that grows by 5% a year multiplies its balance by 1.05 each year, so after 10 years a deposit of 1,000 is worth 1,000 × 1.05¹⁰. The calculator gives 1.05¹⁰ = 1.62889462677744, so the balance is about 1,628.89. The compound interest calculator wraps this same power in a full savings model.

Negative bases and roots

With a whole-number exponent, a negative base simply alternates sign: (−2)³ = −8 and (−2)⁴ = 16. An odd exponent keeps the sign negative, an even exponent makes the result positive.

Type the minus sign into the base box to raise a negative number. Written on paper, −2² is read as −(2²) = −4, whereas (−2)² = 4. The calculator always works on the number in the base box, so a base of −2 and an exponent of 2 gives 4.

A negative base with a fractional exponent, such as (−8)^0.3333333, has no real answer when computed as a general power, so the calculator shows nan. If you need an odd root of a negative number, take the root of the positive number and attach the sign yourself: the cube root of −8 is −(cube root of 8) = −2.

Large results and precision

Computers store most numbers as binary floating-point values with roughly 15 to 17 significant decimal digits. Below 2⁵³ (9,007,199,254,740,992) every whole number is stored exactly. Above that, only some whole numbers can be represented, so the calculator prints the nearest one.

This is why 10²⁵ appears as 10,000,000,000,000,000,905,969,664 rather than a 1 followed by 25 zeros, and why 3³⁹ appears as 4,052,555,153,018,976,256 although the exact value is 4,052,555,153,018,976,267. The leading digits are right; the trailing ones are rounding noise. For exact arithmetic on very large integers, use a tool built for arbitrary-precision numbers.

At the other end, results smaller than about 0.000000000000005 are rounded to 14 decimals and shown as 0. For tiny or huge numbers it is usually clearer to think in scientific notation.

Table of common powers

Powers of 2 are the backbone of computing: 2¹⁰ = 1,024, 2²⁰ = 1,048,576 and 2³² = 4,294,967,296. See the binary calculator for working in base 2.

Where exponents are used

• Growth and decay: interest, population change and radioactive half-life all multiply by the same factor in each period, which is a power of that factor.
• Roots: raising to the power 1/n is the same as taking the n-th root. See the square root, cube root and any root calculators.
• Logarithms: the logarithm answers the opposite question, "what exponent gives this result?". Try the logarithm calculator and the antilog calculator.
• Geometry and physics: area grows with the square of length, volume with the cube, and many forces follow inverse-square laws.