Exponent Calculator
This exponent calculator works out powers with positive, negative, or fractional exponents, nth roots, and scientific notation conversions. Enter a base and exponent to get the result instantly, with the calculation shown step by step.
Exponent Calculator
Enter a base and an exponent. Works with whole numbers, decimals, negative exponents, and fractional exponents (e.g. 0.5 for square root).
| Base | Exponent |
Find the nth root of a number — for example, the square root (n=2) or cube root (n=3).
| Number | Root (n) |
Convert a regular number into scientific notation (a × 10ⁿ), or convert scientific notation back into a regular number.
| Enter a Number (e.g. 6500000 or 0.000034) |
Core rules of exponents for quick reference.
| Rule | Formula | Example |
|---|---|---|
| Product of powers | xᵃ × xᵇ = xᵃ⁺ᵇ | 2³ × 2² = 2⁵ = 32 |
| Quotient of powers | xᵃ ÷ xᵇ = xᵃ⁻ᵇ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (xᵃ)ᵇ = xᵃᵇ | (2³)² = 2⁶ = 64 |
| Power of a product | (xy)ᵃ = xᵃyᵃ | (2×3)² = 2²×3² = 36 |
| Zero exponent | x⁰ = 1 (x ≠ 0) | 5⁰ = 1 |
| Negative exponent | x⁻ⁿ = 1 / xⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | x^(1/n) = ⁿ√x | 8^(1/3) = ∛8 = 2 |
| General fractional | x^(m/n) = ⁿ√(xᵃ) | 8^(2/3) = ∛(8²) = 4 |
About This Calculator
An exponent (or power) tells you how many times to multiply a number by itself. This calculator handles whole number exponents, negative exponents, fractional exponents (which represent roots), nth roots directly, and conversion to and from scientific notation. It’s useful for students working through algebra homework or anyone who needs a quick way to check a power calculation. If you’re working with percentages instead, our Percentage Calculator may be more useful.
What Is an Exponent?
In the expression xⁿ, x is called the base and n is the exponent (or power). It means x multiplied by itself n times. For example, 2⁴ means 2 × 2 × 2 × 2 = 16. Exponents can also be negative, zero, or fractional, each with its own rule for evaluating the expression.
Calculation Formula
Negative exponent: x⁻ⁿ = 1 / xⁿ
Zero exponent: x⁰ = 1 (for x ≠ 0)
Fractional exponent: x^(1/n) = ⁿ√x
Nth root: ⁿ√x = x^(1/n)
Worked Example
Negative exponent rule: 2⁻³ = 1 / 2³
2³ = 8
Result = 1/8 = 0.125
Calculating 27^(1/3):
Fractional exponent means cube root: ∛27
Result = 3, since 3 × 3 × 3 = 27
Common Mistakes to Avoid
- A negative exponent does not make the result negative — it means “take the reciprocal”, so 2⁻² = 1/4, not −4
- x⁰ always equals 1 for any non-zero x, regardless of how large or small x is — this surprises many students
- Even roots (square root, 4th root, etc.) of a negative number aren’t real numbers, while odd roots (cube root, 5th root) of a negative number are real and negative
- (−2)² = 4, but −2² = −4, because the exponent applies only to the 2 unless the negative sign is inside the brackets
- When converting to scientific notation, the coefficient must always be between 1 and 10 — 65 × 10⁵ should be rewritten as 6.5 × 10⁶
Frequently Asked Questions
What does a fractional exponent mean?
A fractional exponent represents a root. x^(1/n) is the nth root of x, and x^(m/n) is the nth root of x raised to the power m (or equivalently, the nth root of x, then raised to m).
Why is any number to the power of 0 equal to 1?
This follows from the quotient rule: xⁿ ÷ xⁿ = xⁿ⁻ⁿ = x⁰, and anything divided by itself equals 1, so x⁰ must equal 1 for the rule to stay consistent.
Can you take an even root of a negative number?
Not within real numbers. The square root, 4th root, or any even root of a negative number is undefined in the real number system and requires imaginary numbers to express. Odd roots of negative numbers, like the cube root of −8 (which is −2), are perfectly normal real numbers.
How is scientific notation different from a regular exponent calculation?
Scientific notation is a way of writing very large or very small numbers using a power of 10, in the form a × 10ⁿ where 1 ≤ a < 10. It uses exponents but is really a notation convention rather than a calculation rule on its own.
Does this calculator handle imaginary numbers?
No. This calculator only returns real number results. Even roots of negative numbers, or other operations that would require imaginary numbers, will show as undefined.
This exponent calculator is provided for educational and reference purposes only. Results are limited to real numbers; calculations that would require imaginary numbers (such as even roots of negative numbers) are not supported.