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Formula
bⁿ = b × b × … × b (n times); b⁻ⁿ = 1 ÷ bⁿ; b^(m/n) = (ⁿ√b)ᵐ
- b
- the base, the number being multiplied
- n
- the exponent (power), how many times the base is used as a factor
- m/n
- a fractional exponent, where n is the root and m is the power
What an exponent means
An exponent tells you how many times to use a number as a factor. In 5⁴, the base is 5 and the exponent (or power) is 4, so 5⁴ = 5 × 5 × 5 × 5 = 625. You read it as “five to the fourth power”. The second and third powers have their own names, squared and cubed, because they give the area of a square and the volume of a cube.
Exponents aren’t limited to positive whole numbers. Negative exponents stand for reciprocals, and fractional exponents stand for roots. The exponent calculator handles all of these and says plainly when a power has no real value.
How to use the exponent calculator
- Base and exponent tab: enter the base and the exponent. The exponent can be a whole number, a negative number or a decimal such as 0.5.
- Fractional exponent tab: enter the base, then the numerator m and denominator n of the exponent. Use this tab for powers like 8^(2/3), and for odd roots of negative numbers such as (−64)^(1/3), which a decimal exponent can’t express exactly.
- The answer updates as you type. Very large or very small results switch to scientific notation. For whole-number powers, the exact value with every digit appears underneath.
- The steps show the repeated multiplication, the reciprocal for negative exponents, or the root and then the power for fractional exponents.
How to calculate exponents by hand
Whole-number exponents
Multiply step by step: 4⁵ = 4 × 4 × 4 × 4 × 4 = 1,024. For larger exponents, square repeatedly. 7⁸ = ((7²)²)² = (49²)² = 2,401² = 5,764,801. That’s three multiplications instead of seven.
Negative exponents
Flip to the reciprocal, then use the positive power. 2⁻⁵ = 1 ÷ 2⁵ = 1/32 = 0.03125. A negative exponent on a fraction flips the fraction: (2/3)⁻² = (3/2)² = 9/4 = 2.25.
Fractional exponents
Take the root given by the denominator, then raise it to the numerator. 32^(3/5): the fifth root of 32 is 2, and 2³ = 8. Taking the root first keeps the numbers small, although (32³)^(1/5) gives the same answer.
Exponent rules
| Rule | Example |
|---|---|
| bᵐ × bⁿ = bᵐ⁺ⁿ | 3² × 3⁴ = 3⁶ = 729 |
| bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 5⁷ ÷ 5⁴ = 5³ = 125 |
| (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 |
| (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| b⁰ = 1 (b ≠ 0) | 9⁰ = 1 |
| b⁻ⁿ = 1 ÷ bⁿ | 10⁻³ = 0.001 |
| b^(1/n) = ⁿ√b | 81^(1/4) = 3 |
These rules apply only when the bases match (for the first three) or the exponents match (for the fourth). 2³ × 3² can’t be combined into a single power.
Powers of 2 and 10
| n | 2ⁿ | 10ⁿ |
|---|---|---|
| −3 | 0.125 | 0.001 |
| −1 | 0.5 | 0.1 |
| 0 | 1 | 1 |
| 1 | 2 | 10 |
| 5 | 32 | 100,000 |
| 8 | 256 | 100,000,000 |
| 10 | 1,024 | 10,000,000,000 |
| 16 | 65,536 | 10¹⁶ |
| 32 | 4,294,967,296 | 10³² |
Powers of 2 count computer memory and binary values, while powers of 10 are the basis of scientific notation and metric prefixes.
Where exponents are used
- Compound growth. Money growing at 5% a year for 10 years is multiplied by 1.05¹⁰ ≈ 1.6289. Population, bacteria and viral sharing follow the same pattern.
- Decay. A quantity that halves every period keeps (1/2)ⁿ of its original amount after n periods.
- Area and volume. A square’s area is s², and a cube’s volume is s³. Doubling the side quadruples the area and multiplies the volume by eight.
- Scientific notation. The speed of light is about 3 × 10⁸ m/s, and a hydrogen atom is about 10⁻¹⁰ m across.
- Prime factorizations. 360 = 2³ × 3² × 5. These exponents drive the LCM calculator and GCF calculations.
- Roots. A square root is the exponent 1/2. The square root calculator also gives the simplified radical form.
Common mistakes
- Multiplying the base by the exponent. 3⁴ is 81, not 12.
- Reading a negative exponent as a negative answer. 4⁻² = 1/16, which is positive.
- Dropping the parentheses on a negative base. −2⁴ = −16, but (−2)⁴ = 16. The calculator shows parentheses so it’s clear which one you mean.
- Adding exponents on different bases. 2³ × 3² is 8 × 9 = 72, not 6⁵.
- Raising a sum term by term. (a + b)² = a² + 2ab + b², not a² + b². Expressions like these lead to quadratic equations, which the quadratic formula calculator solves.
Glossary
- Base: the number being multiplied.
- Exponent (power, index): how many times the base is used as a factor.
- Reciprocal: 1 divided by a number. The reciprocal of 8 is 1/8.
- Rational exponent: an exponent written as a fraction, m/n.
Worked examples
Whole-number exponent
Raise 3 to the 5th power.
Inputs (Base and exponent): Base (b) = 3; Exponent (n) = 5
3⁵ = 3 × 3 × 3 × 3 × 3 3⁵ = 243
Answer: 3⁵ = 243
Negative exponent
A negative exponent turns the power into a reciprocal: 4⁻³ = 1 ÷ 4³.
Inputs (Base and exponent): Base (b) = 4; Exponent (n) = -3
A negative exponent means a reciprocal: 4⁻³ = 1 ÷ 4³ 4³ = 64 4⁻³ = 0.015625
Answer: 4⁻³ = 0.015625
Fractional exponent of a negative base
Edge case: (−27)^(2/3) is real because the cube root of a negative number exists.
Inputs (Fractional exponent): Base (b) = -27; Exponent numerator (m) = 2; Exponent denominator (n) = 3
b^(m/n) = (ⁿ√b)ᵐ, so (−27)^(2/3) = (∛(−27))² ∛(−27) = −3 (−3)² = 9 (−27)^(2/3) = 9
Answer: (−27)^(2/3) = 9
Doubling 64 times
A quantity that doubles 64 times grows by a factor of 2⁶⁴. The exact value is shown too.
Inputs (Base and exponent): Base (b) = 2; Exponent (n) = 64
2⁶⁴ = 2 multiplied by itself 64 times 2⁶⁴ = 1.84467 × 10¹⁹
Answer: 2⁶⁴ = 1.84467 × 10¹⁹
Assumptions and limitations
- The base can be any number from −1,000,000,000,000,000 to 1,000,000,000,000,000. The exponent can be from −10,000 to 10,000 in the base-and-exponent tab. In the fractional tab, the numerator can be from −1,000 to 1,000 and the denominator from 1 to 1,000.
- Whole-number powers of whole numbers are calculated exactly, up to 300 digits. When the result has 16 or more digits, the main answer is shown in scientific notation and the exact digits are listed underneath.
- Other results are shown to up to 10 decimal places. Values smaller than 0.0001 or of 10¹⁵ and above use scientific notation (× 10ⁿ). Results beyond the normal floating-point range are found with logarithms, to 6 significant digits.
- 0 raised to a negative power is undefined, because it means dividing by 0.
- A negative base with a decimal exponent has no real value. Enter it as a fraction in the Fractional exponent tab instead. An odd root, such as (−8)^(1/3), is real, while an even root of a negative number isn't.
- Fractional exponents are reduced to lowest terms before evaluating, so (−8)^(2/6) is treated as (−8)^(1/3) = −2.
- 0⁰ is given as 1, the convention in algebra and combinatorics, with a note that calculus treats it as an indeterminate form.
Frequently asked questions
How do you calculate exponents?
Multiply the base by itself as many times as the exponent says. 6³ = 6 × 6 × 6 = 216. For larger powers, square repeatedly. 3⁸ = ((3²)²)² = (9²)² = 81² = 6,561.
How do you calculate negative exponents?
A negative exponent means the reciprocal of the positive power. 5⁻² = 1 ÷ 5² = 1/25 = 0.04. The sign of the exponent never makes the answer negative; only a negative base can do that.
What is a fractional exponent?
The denominator is a root and the numerator is a power. 32^(3/5) means take the fifth root of 32 (which is 2) and cube it, giving 8. An exponent of 1/2 is the same as a square root, so 49^(1/2) = 7.
Why is a negative number to a fractional power sometimes undefined?
An even root of a negative number isn't real. No real number squared gives −16, so (−16)^(1/2) is undefined in the real numbers. An odd root is fine. (−125)^(1/3) = −5, because (−5)³ = −125.
Why is anything to the power of 0 equal to 1?
Each step down in the exponent divides by the base. 2³ = 8, 2² = 4, 2¹ = 2, so 2⁰ = 1. The same pattern gives 2⁻¹ = 1/2. This works for any non-zero base.
How do I enter exponents on a scientific calculator?
Look for a key labeled xʸ, yˣ, ^ or xⁿ. Type the base, press that key, then type the exponent. For a negative base, put it in parentheses, because −3^2 is read as −(3²) = −9, while (−3)^2 = 9.
How do I see a large power without scientific notation?
For whole-number bases and exponents, the calculator lists the exact value with every digit, up to 300 digits, below the main answer. 3⁴⁰, for example, is shown in full as a 20-digit number.
Sources
- 1.2 Exponents and Scientific Notation — College Algebra 2e — OpenStax, Rice University. Checked October 10, 2026.
- 1.3 Radicals and Rational Exponents — College Algebra 2e — OpenStax, Rice University. Checked October 10, 2026.
- Exponentiation — Wolfram MathWorld. Checked October 10, 2026.