ⁿ Exponent Calculator
Raise any number to any power – including negative and fractional exponents – or take an nth root, with the exponent rules applied step by step.
Quick answer: 8^(2/3) means take the cube root of 8 and square it: (∛8)² = 2² = 4; a negative exponent flips the result, so 8^(−2/3) = 1/4 = 0.25.
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Exponent Calculator inputs
Result
8^(2/3) =
4
- Scientific notation
- 4 × 10⁰
- log₁₀ of the result
- 0.60206
Step by step
- Fractional exponent 2/3: take the cube root, then raise to the power 2
- ∛8 = 2
- 2² = 4
- As a decimal exponent: 8^0.66666667 = 4
Exponent rules
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ = 729 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1 ÷ aⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | a^(m/n) = (ⁿ√a)ᵐ | 8^(2/3) = (∛8)² = 4 |
Powers of 8
| Power | Value |
|---|---|
| 8⁻³ | 0.001953125 |
| 8⁻² | 0.015625 |
| 8⁻¹ | 0.125 |
| 8⁰ | 1 |
| 8¹ | 8 |
| 8² | 64 |
| 8³ | 512 |
| 8⁴ | 4,096 |
| 8⁵ | 32,768 |
| 8¹⁰ | 1,073,741,824 |
What an exponent means
An exponent tells you how many times to multiply the base by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The rules below extend this to zero, negative and fractional exponents.
aⁿ = a × a × … × a (n times) · a⁻ⁿ = 1 ÷ aⁿ · a^(m/n) = (ⁿ√a)ᵐ
Worked example
Evaluate 8^(2/3):
- The denominator 3 means a cube root: ∛8 = 2.
- The numerator 2 means square it: 2² = 4.
- So 8^(2/3) = 4. With a negative exponent, 8^(−2/3) = 1 ÷ 4 = 0.25.
The exponent rules
| Rule | Formula |
|---|---|
| Product rule | aᵐ · aⁿ = aᵐ⁺ⁿ |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power rule | (aᵐ)ⁿ = aᵐⁿ |
| Zero exponent | a⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ |
How to use the calculator
Enter the base and the exponent. The exponent can be a whole number (3), a negative number (−2), a decimal (0.5) or a fraction (2/3). For roots, switch to “nth root” and enter the number and the root index. The working shows how each rule is applied.
Roots as exponents
Every root is a fractional power: √x = x^(1/2), ∛x = x^(1/3) and the nth root is x^(1/n). Choose “nth root” above to find, for example, the 5th root of 32 (= 2). Odd roots of negative numbers are real; even roots of negative numbers are not.
Big results
When the base and exponent are whole numbers the calculator also computes the exact integer, however many digits it has – 2^100 = 1,267,650,600,228,229,401,496,703,205,376 – instead of rounding to scientific notation.
Frequently asked questions
What does a negative exponent mean?
A negative exponent means “one divided by” the positive power: a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ = 1 ÷ 2³ = 1/8 = 0.125. It does not make the result negative.
How do fractional exponents work?
The denominator is a root and the numerator is a power: a^(m/n) = (ⁿ√a)ᵐ. 27^(2/3) = (∛27)² = 3² = 9, and 16^(1/2) is simply √16 = 4.
Why is any number to the power 0 equal to 1?
Dividing powers subtracts exponents: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and anything divided by itself is 1. The case 0⁰ is left undefined here, although many fields use 1 by convention.
Can a negative number have a fractional exponent?
Only if the root is odd. (−8)^(1/3) = −2 because ∛(−8) = −2, but (−4)^(1/2) = √(−4) has no real value – it is the imaginary number 2i.