How to use the Exponent Calculator
- Enter the base, the number you want to multiply by itself.
- Enter the exponent, which can be negative or a decimal.
- Read the result, its scientific notation and the reciprocal.
- Try fractional exponents such as 0.5 to find roots.
Formula used
xⁿ means x multiplied by itself n times. x⁰ = 1, x⁻ⁿ = 1 ÷ xⁿ, and x^(1/n) is the nth root of x.
Worked example
Here is the Exponent Calculator worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Base (x) | 2 |
| Exponent (n) | 10 |
| Result | |
| Result (xⁿ) | 1,024 |
| Scientific notation | 1.024 × 10^3 |
| Reciprocal (x⁻ⁿ) | 0.0009765625 |
What exponents mean
In 2¹⁰ the base is 2 and the exponent is 10, so the result is 2 multiplied by itself 10 times: 1,024. Exponents grow quickly: 2²⁰ is 1,048,576, and 2¹⁰⁰ is 1.267651 × 10^30.
Any number to the power of 0 is 1, so 1 for 7⁰. Any number to the power of 1 is itself. A negative base gives a positive result for an even exponent, as in (−3)² = 9, and a negative result for an odd exponent, as in (−2)³ = -8.
Negative exponents
A negative exponent means the reciprocal: x⁻ⁿ = 1 ÷ xⁿ. So 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125, and 10⁻² = 0.01. That is why the calculator also shows the reciprocal of the result, which is the same as flipping the sign of the exponent.
Negative exponents are used to write very small numbers, such as the width of a human hair, about 7 × 10⁻⁵ metres.
Fractional exponents and roots
A fractional exponent is a root. x^(1/2) is the square root, and x^(1/3) is the cube root. So 9^0.5 = 3, and 27^(1/3) is about 3. In general x^(m/n) is the nth root of x raised to the power m.
A negative base with a fractional exponent usually has no real-number result, such as (−4)^0.5, because it would require the square root of a negative number. The calculator tells you when that happens.
Scientific notation and large numbers
Scientific notation writes a number as a coefficient times a power of ten. The result of 2¹⁰⁰ is 1.267651 × 10^30. Results above one quadrillion (10¹⁵) or below one millionth are displayed this way, and the largest value the calculator can display is about 1.8 × 10³⁰⁸.
Exponents are used in compound interest, population growth, computing (powers of 2), physics and chemistry. The compound interest calculator on this site is one application, and the Rule of 72 calculator is another.
Laws of exponents and powers of 2
Three rules make exponents easier. When you multiply powers with the same base, add the exponents: 2³ × 2⁴ = 2⁷ = 128. When you divide, subtract them: 2⁷ ÷ 2³ = 2⁴ = 16. When you raise a power to a power, multiply them: (2³)² = 2⁶ = 64.
Powers of 2 appear everywhere in computing: 2⁸ = 256, 2¹⁰ = 1,024, 2¹⁶ = 65,536 and 2³² = 4,294,967,296. Powers of 10 give scientific notation: 10⁶ is a million and 10⁹ a billion. Remember that (a + b)ⁿ is not aⁿ + bⁿ.
Frequently asked questions
What is a negative exponent?
It means the reciprocal. 2⁻³ is 1 ÷ 2³ = 1/8 = 0.125.
What is a fractional exponent?
It is a root. x^(1/2) is the square root of x and x^(1/3) is the cube root.
What is anything to the power of zero?
One, for any non-zero base. The value of 0⁰ is debated, but it is often defined as 1.
Why does a negative base with a decimal exponent fail?
It would need roots of negative numbers, which are not real numbers.
How big a result can it show?
Up to about 1.8 × 10³⁰⁸. Larger results are reported as too large.
How do I calculate 10 to a power?
Enter 10 as the base. 10³ is 1,000 and 10⁻³ is 0.001.
How do I multiply numbers with exponents?
If the bases are the same, add the exponents. 2³ × 2⁴ = 2⁷. If the bases differ, calculate each power first.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.