Exponent Calculator ($b^n$)
Calculate the power of a number instantly. Enter the Base ($b$) and the Exponent ($n$) below. Our calculator handles integers, decimals, fractions, and negative exponents, and provides a clear step-by-step breakdown.
Calculate: $b^n$
Result
Final Value
Step-by-Step Breakdown
The Laws of Exponents
Exponentiation is governed by a few fundamental rules, or laws, that simplify multiplication and division of powers.
| Law | Formula | Example |
|---|---|---|
| Product Rule | $b^m \cdot b^n = b^{m+n}$ | $2^3 \cdot 2^2 = 2^5 = 32$ |
| Quotient Rule | $$\frac{b^m}{b^n} = b^{m-n}$$ | $$\frac{3^5}{3^2} = 3^3 = 27$$ |
| Power Rule | $(b^m)^n = b^{m \cdot n}$ | $(4^2)^3 = 4^6 = 4096$ |
| Negative Exponent Rule | $$b^{-n} = \frac{1}{b^n}$$ | $$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$ |
| Zero Exponent Rule | $b^0 = 1, \text{ for } b \neq 0$ | $78^0 = 1$ |
Fractional and Decimal Exponents
When the exponent is a fraction or decimal, it relates directly to taking a root. A fractional exponent $n = \frac{p}{q}$ is handled by the Root Law:
$$b^{p/q} = \sqrt[q]{b^p} = (\sqrt[q]{b})^p$$
The denominator, $q$, determines the root (e.g., square root, cube root), and the numerator, $p$, determines the power.
Example: $8^{2/3}$ is the cube root of $8$ (which is $2$), raised to the power of $2$ (which is $4$).