Exponent Calculator

What is an exponent?

An exponent is the small raised number in a power such as 5³. It counts how many times the base, the large number, is multiplied by itself: 5³ = 5 × 5 × 5 = 125. The exponent is also called the power or the index, and 5³ is read "5 to the power of 3", "5 to the third" or "5 cubed". A power of 2 is called a square, so 5² is "5 squared".

How to calculate exponents

For a whole-number exponent, write the base down that many times and multiply from left to right. 2¹⁰ is 2 × 2 = 4, × 2 = 8, and so on up to 1,024. For bigger exponents, square repeatedly, which saves a lot of steps: 3² = 9, (3²)² = 3⁴ = 81, (3⁴)² = 3⁸ = 6,561, three multiplications instead of seven.

Show 3⁸ in the calculator

On a phone or scientific calculator, use the key marked xʸ, yˣ or ^. In Excel and Google Sheets type =2^10 or =POWER(2, 10). Large answers come back in scientific notation, so 2¹⁰⁰ shows as about 1.27E+30. This calculator keeps all 31 digits.

Exponent rules

The laws of exponents let you simplify powers without working them out. Most of them need the same base. The power of a product and the power of a quotient work the other way round, with different bases under one exponent. Each example opens in the calculator.

Three mistakes come up again and again:

• Adding powers does not add exponents. 2³ + 2⁴ = 8 + 16 = 24, which is not 2⁷ = 128. The product rule is for multiplying.
• A power of a sum is not the sum of powers. (3 + 4)² = 49, but 3² + 4² = 25.
• Multiplying the base by the exponent. 2⁵ is 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10.

Negative exponents

A negative exponent means one over the positive power: a⁻ⁿ = 1/aⁿ. So 2⁻³ = 1/2³ = 1/8 = 0.125. The answer gets smaller, but it does not turn negative. One way to see why: each step down in the exponent divides by the base. 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4, 2⁻³ = 1/8.

Show 2⁻³ in the calculator

With a fraction as the base, the quickest way by hand is to flip the fraction and drop the minus sign: (2/3)⁻² = (3/2)² = 9/4.

Zero exponent

Any number other than 0 to the power of 0 is 1. The quotient rule shows why: 5³ ÷ 5³ = 5³⁻³ = 5⁰, and anything divided by itself is 1. The case 0⁰ is different. Algebra, counting and most calculators, this one included, take it as 1, but in calculus it is left undefined, because the limit depends on how both parts approach 0.

Fractional exponents

A fractional (rational) exponent combines a root and a power. In a^p/q, the bottom number q is the root and the top number p is the power. Take the root first, because it keeps the numbers small: 27^2/3 is the cube root of 27, which is 3, then 3² = 9. An exponent of 1/2 is a square root, 1/3 a cube root, and so on.

Show 27^2/3 in the calculator

When the root does not come out even, the exact answer keeps a root sign. 8^1/2 = √8 = 2√2, about 2.828, because 8 = 4 · 2 and √4 = 2 comes out. Decimal exponents are fractions too: 16^0.75 = 16^3/4 = 2³ = 8. Typing the fraction gives a cleaner answer than typing a rounded decimal such as 0.333 for 1/3, which only gets close.

Show 8^1/2 = 2√2

Negative bases and brackets

Brackets decide what the exponent applies to. (−2)⁴ multiplies −2 four times and gives 16, while −2⁴ means −(2⁴) = −16. An odd power of a negative number stays negative: (−2)³ = −8.

Odd roots of negative numbers are fine, so (−8)^1/3 = −2. Even roots are not: no real number times itself gives −4, so (−4)^1/2 has no real value. In complex numbers it is 2i, and the calculator says so instead of showing an error.

Show (−8)^1/3 in the calculator

Large numbers and scientific notation

Powers grow fast. 2¹⁰⁰ has 31 digits, so it is easier to read in scientific notation, a number from 1 to 10 times a power of 10: about 1.2677 × 10³⁰. The power of 10 tells you the size at a glance. To count the digits of bⁿ without working it out, take n × log₁₀ b, round down and add 1. For 2¹⁰⁰: 100 × 0.30103 = 30.1, so 31 digits.

Show every digit of 2¹⁰⁰

Powers of 2

Each row doubles the one before. Computers count in powers of 2, so 2¹⁰ = 1,024, 2¹⁶ = 65,536 and 2²⁰ = 1,048,576 turn up in memory sizes and colour depths.

Powers of 10

A positive power of 10 is a 1 followed by that many zeros. A negative power moves the 1 that many places to the right of the decimal point.

Solving for the exponent

Sometimes you know the base and the answer and need the exponent: 2 to what power is 1,024? Logarithms undo powers, so x = log(1024) ÷ log(2) = 10. Any log works, as long as you use the same one on top and bottom. Switch the calculator to Find the exponent to do this. It gives an exact answer when there is one (8ˣ = 4 gives x = 2/3) and a rounded one otherwise (2ˣ = 10 gives x ≈ 3.322).

Solve 2ˣ = 1024 in the calculator

Where exponents are used

Compound interest is a power: $1,000 at 5% a year for 10 years grows to 1,000 × 1.05¹⁰, about $1,628.89. Area and volume use squares and cubes, which is where "squared" and "cubed" come from: a square with 5 cm sides covers 5² = 25 cm². Science writes very large and very small quantities as powers of 10, and computer memory is counted in powers of 2.

Show 1.05¹⁰ in the calculator