Exponent Calculator
Raise any number to a power, including negative and fractional exponents, get exact and decimal answers, or solve for a missing exponent or base, with every step shown.
- Expression
- 2⁵
- Base
- 2
- Exponent
- 5
- Exact result
- 32
- Decimal result
- 32
- Rule used
- aⁿ = a × a × … × a (n factors)
Step-by-step solution
Step 1: Multiply the base by itself
2⁵ = 2 × 2 × 2 × 2 × 2
= 32
Step 2: Answer
2⁵ = 32
How the powers of 2 grow
| 2^1 | 2 | |
|---|---|---|
| 2^2 | 4 | |
| 2^3 | 8 | |
| 2^4 | 16 | |
| 2^5 | 32 |
Bars use a logarithmic scale (number of digits), because the values grow too fast to compare on a normal scale: each extra power multiplies the value by 2.
Related tools: undo a power with the Logarithm Calculator, write very large or small powers of 10 with the Scientific Notation Calculator, work with fractional bases in the Fraction Calculator, and evaluate longer expressions in the Scientific Calculator.
An exponent tells you how many times to multiply a number by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. This exponent calculator raises any base to any power, whether the exponent is positive, zero, negative (2⁻³ = 1/8), a fraction (16^(3/2) = 64) or a decimal. It keeps answers exact where possible, such as 1/8 or 2√2, alongside the decimal value.
You can also type expressions like −2^2 and (−2)^2 to see why they differ, solve aˣ = b for a missing exponent, or solve xⁿ = y for a missing base. Every result includes a step-by-step solution built from your numbers.
What Is an Exponent?
An exponent says how many times to multiply a number by itself. In 5³, 5 is the base (the number being multiplied) and 3 is the exponent (how many factors): 5³ = 5 × 5 × 5 = 125. It is read “5 to the power of 3” or “5 cubed”.
The word power is used for the whole expression or its value: 125 is the third power of 5. The exponent is just the small raised number. On a keyboard, exponents are written with a caret: 5^3.
Exponent Formula
aⁿ = a × a × … × a (n factors)
Here a is the base and n the exponent. The meaning extends beyond whole numbers in a way that keeps the exponent rules working:
| Exponent | Meaning | Example |
|---|---|---|
| Positive whole number | Repeated multiplication | 3⁴ = 3 × 3 × 3 × 3 = 81 |
| Zero | Always 1 (for a ≠ 0) | 7⁰ = 1 |
| Negative | Reciprocal of the positive power | 2⁻³ = 1/2³ = 1/8 |
| Fraction m/n | n-th root, raised to the power m | 8^(2/3) = (∛8)² = 4 |
| Decimal | The same as the equivalent fraction | 9^0.5 = 9^(1/2) = 3 |
Exponent Rules
These laws of exponents apply when the bases (or the exponents) match. They follow directly from counting factors: 2³ × 2⁴ is 3 twos times 4 twos, which is 7 twos.
Product of powers
aᵐ × aⁿ = aᵐ⁺ⁿ
2³ × 2⁴ = 2⁷ = 128
Quotient of powers
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
5⁶ ÷ 5⁴ = 5² = 25
a ≠ 0
Power of a power
(aᵐ)ⁿ = aᵐⁿ
(3²)³ = 3⁶ = 729
Safe for whole-number exponents; with fractional exponents and negative bases it can fail, e.g. ((−1)²)^(1/2) = 1, not −1.
Power of a product
(ab)ⁿ = aⁿbⁿ
(2 × 5)³ = 2³ × 5³ = 1,000
Power of a quotient
(a/b)ⁿ = aⁿ / bⁿ
(2/3)² = 4/9
b ≠ 0
Zero exponent
a⁰ = 1
7⁰ = 1
a ≠ 0; 0⁰ is left undefined here
Negative exponent
a⁻ⁿ = 1 / aⁿ
2⁻³ = 1/8
a ≠ 0
Fractional exponent
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
8^(2/3) = (∛8)² = 4
For even n, a must be ≥ 0 to stay in the real numbers.
There is no rule for sums: (a + b)² ≠ a² + b². For example (2 + 3)² = 25, but 2² + 3² = 13.
Negative Exponents
a⁻ⁿ = 1 / aⁿ
A negative exponent means “divide instead of multiply”. Each step down in the exponent divides by the base: 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = 1/2, 2⁻² = 1/4. So 3⁻² = 1/3² = 1/9 ≈ 0.1111. A negative exponent does not make the answer negative, and a fraction with a negative exponent flips: (2/3)⁻² = (3/2)² = 9/4. Zero has no negative powers, because 0⁻¹ = 1/0 is undefined.
Fractional Exponents and Roots
a^(1/n) = ⁿ√a a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
The denominator of a fractional exponent is the root and the numerator is the power. Taking the root first usually keeps the numbers smaller: 16^(3/2) = (√16)³ = 4³ = 64. When the root isn’t a whole number, the exact answer is a radical, such as 2^(1/2) = √2 ≈ 1.41421356 or 2^(3/2) = 2√2.
| Root form | Exponent form | Value |
|---|---|---|
| √25 | 25^(1/2) | 5 |
| ∛27 | 27^(1/3) | 3 |
| ∜16 | 16^(1/4) | 2 |
| √(4³) = (√4)³ | 4^(3/2) | 8 |
| 1 / √9 | 9^(−1/2) | 1/3 |
Negative bases: odd roots of negative numbers are real ((−8)^(1/3) = −2), but even roots are not, because no real number squared is negative. So (−4)^(1/2) has no real value; this calculator explains that instead of returning a complex number.
How Exponents Grow
Increasing the exponent by 1 multiplies the value by the base, so powers grow much faster than multiples. Compare doubling with adding 2:
| n | 2 × n (linear) | 2ⁿ (exponential) |
|---|---|---|
| 1 | 2 | 2 |
| 2 | 4 | 4 |
| 3 | 6 | 8 |
| 4 | 8 | 16 |
| 5 | 10 | 32 |
| 10 | 20 | 1,024 |
| 20 | 40 | 1,048,576 |
By n = 20 the linear sequence has reached 40, while 2²⁰ is over a million. For a base between 0 and 1, the opposite happens: each higher power is smaller (0.5³ = 0.125).
Exponent Examples
Compound growth
A value of 1,000 grows 10% a year for 3 years
Final = Start × (1 + r)ⁿ
1.1³ = 1.1 × 1.1 × 1.1 = 1.331
1,000 × 1.331 = 1,331
1,331 after 3 years
Growing by the same percentage each period is repeated multiplication, so the exponent counts the periods. The Compound Interest Calculator handles deposits and compounding frequency.
Area and volume
A square with 4 cm sides; a cube with 4 cm edges
Area = s², Volume = s³
4² = 4 × 4 = 16
4³ = 4 × 4 × 4 = 64
16 cm² and 64 cm³
That is why area is in square units (cm²) and volume in cubic units (cm³): the exponent matches the number of dimensions.
Computer storage
Binary: each bit doubles the number of values
2ⁿ values from n bits
2⁸ = 256
2¹⁰ = 1,024
1 byte = 256 values; 1 KiB = 1,024 bytes
Memory sizes are powers of 2, which is why a kibibyte is 1,024 bytes rather than 1,000.
Scientific notation
One million, and the distance light travels in a year
10ⁿ = 1 followed by n zeros
10⁶ = 1,000,000
9.46 × 10¹⁵ m ≈ 9,460,000,000,000,000 m
10⁶ = 1,000,000
Powers of 10 make very large and very small numbers readable. The Scientific Notation Calculator converts and calculates with them.
Negative powers of 10
A millimetre in metres
10⁻ⁿ = 1 / 10ⁿ
10⁻³ = 1 / 10³ = 1 / 1,000
= 0.001
1 mm = 10⁻³ m = 0.001 m
A negative exponent does not make the number negative: it makes it a fraction. Each step down divides by 10.
For growth with regular deposits, use the Compound Interest Calculator; for powers of 10, the Scientific Notation Calculator.
Common Exponent Mistakes
- −2² vs (−2)²: the power is applied before the minus sign, so −2² = −(2²) = −4, while (−2)² = (−2) × (−2) = 4. Calculators and spreadsheets follow the same order, so use parentheses when the base is negative.
- Negative exponent vs negative number: 2⁻² is 1/4, a positive number. It is not −4.
- Multiplying instead of repeating: 3⁴ is 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12.
- Missing parentheses on fractional exponents: type 9^(1/2). Without them, 9^1/2 means 9¹ ÷ 2 = 4.5.
- Adding exponents of different bases: 2³ × 3² is not 6⁵; the product rule only works when the bases are the same.
- 0⁰: it is not simply 1 in every context. This calculator leaves it undefined and explains why.
How to Use the Exponent Calculator
- Choose a mode: a power, a negative or fractional exponent, a typed expression, or solve for a missing exponent (aˣ = b) or base (xⁿ = y).
- Enter the base and exponent. Negative numbers, decimals (1.5) and fractions (3/4) are accepted; for a fractional exponent, enter its numerator and denominator.
- In Expression mode, type expressions like 2^5, (-2)^3, -2^2 or 9^(1/2). A minus sign without parentheses is applied after the power.
- Read the exact result (such as 1/8 or 2√2) and the decimal result; choose how many decimal places to show. Large and small results also appear in scientific notation.
- Follow the step-by-step solution, which uses your numbers, to see how the power, root or reciprocal was worked out.
Frequently Asked Questions
What is an exponent?
An exponent is the small raised number that says how many times to multiply the base by itself. In 5³, the exponent 3 means 5 × 5 × 5 = 125.
How do you calculate a power?
Multiply the base by itself as many times as the exponent says. 3⁴ = 3 × 3 × 3 × 3 = 81. For large exponents, a calculator is the practical way; this one also shows the result in scientific notation.
What happens when an exponent is zero?
Any non-zero number raised to the power 0 equals 1, because aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ and anything divided by itself is 1. 0⁰ is an indeterminate form and is left undefined here.
What is a negative exponent?
A negative exponent means the reciprocal of the positive power: a⁻ⁿ = 1/aⁿ. For example 2⁻³ = 1/2³ = 1/8 = 0.125. It makes the value smaller, not negative.
How do you calculate a fractional exponent?
Take the root given by the denominator and the power given by the numerator: a^(m/n) = (ⁿ√a)ᵐ. For 16^(3/2), √16 = 4 and 4³ = 64.
What is the difference between a base and an exponent?
The base is the number being multiplied and the exponent is how many times it is used as a factor. In 2⁵, 2 is the base and 5 is the exponent.
Why is (−2)² different from −2²?
By the order of operations, exponents come before the minus sign. −2² means −(2²) = −4, while (−2)² means (−2) × (−2) = 4. Use parentheses to raise a negative number to a power.
What does a fractional exponent mean?
It means a root. An exponent of 1/2 is a square root, 1/3 a cube root, and 1/n an n-th root; a numerator other than 1 adds a power, so 8^(2/3) = (∛8)² = 4.
How are exponents related to roots?
Roots are fractional exponents: √a = a^(1/2) and ∛a = a^(1/3). Roots undo powers, so √(5²) = 5 and ∛(2³) = 2 for positive numbers.
Can an exponent be a decimal?
Yes. A decimal exponent equals a fraction, so 9^0.5 = 9^(1/2) = 3 and 2^2.5 = 2^(5/2) = 4√2 ≈ 5.657. Decimal exponents of negative bases often have no real value.
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Last updated: September 29, 2026.