Exponent Calculator
Calculate powers, exponents, and roots with step-by-step solutions. Free online exponent calculator with scientific notation support.
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Exponent Calculator
This free online exponent calculator computes powers ($a^n$), nth roots ($\sqrt[n]{a}$), and converts between decimal and scientific notation, all in your browser with no signup required.
- Positive integer exponents: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$ (step-by-step expansion shown) - Negative exponents: $2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$ - Fractional exponents: $8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$
$$a^m \times a^n = a^{m+n}$$ $$\frac{a^m}{a^n} = a^{m-n}$$ $$(a^m)^n = a^{m \times n}$$ $$(ab)^n = a^n \times b^n$$ $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$ $$a^{-n} = \frac{1}{a^n}$$ $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$ $$a^0 = 1 \quad (a \neq 0)$$
- Compound Interest: $A = P(1 + r)^t$ — exponential growth in finance - Population Growth: $P(t) = P_0 e^{rt}$ — biological and demographic modeling - Scientific Measurement: pH scale ($-\log[H^+]$), Richter scale, decibels - Computer Science: Binary representation, algorithm complexity ($O(n^2)$, $O(2^n)$) - Physics: Radioactive decay, inverse-square laws, wave functions - Engineering: Signal processing, exponential smoothing, error analysis
Frequently Asked Questions
Got questions about exponents, powers, or how to use this calculator? We've got answers.
What is an exponent?
An exponent (or power) indicates how many times a number (the base) is multiplied by itself. In $a^n$, $a$ is the base and $n$ is the exponent. For example, $2^3 = 2 \times 2 \times 2 = 8$.
How do negative exponents work?
A negative exponent means the reciprocal of the base raised to the positive exponent: $a^{-n} = \frac{1}{a^n}$ (for $a \neq 0$). For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$.
What about fractional exponents?
A fractional exponent like $a^{m/n}$ means the nth root of $a^m$: $a^{m/n} = \sqrt[n]{a^m}$. For example, $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$. The numerator indicates a power and the denominator indicates a root.
What is scientific notation?
Scientific notation expresses numbers as $a \times 10^b$, where $1 \leq |a| < 10$ and $b$ is an integer. For example, $1234 = 1.234 \times 10^3$ and $0.00056 = 5.6 \times 10^{-4}$. It simplifies working with very large or very small numbers.
How do I calculate the nth root?
The nth root of a number $a$, written $\sqrt[n]{a}$, is the value that when raised to the power of $n$ gives $a$. It is equivalent to $a^{1/n}$. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$.
What are the basic exponent rules?
The key exponent rules are: $a^m \times a^n = a^{m+n}$ (product rule), $a^m / a^n = a^{m-n}$ (quotient rule), $(a^m)^n = a^{m \times n}$ (power rule), $(ab)^n = a^n \times b^n$ (power of a product), and $a^0 = 1$ (for $a \neq 0$).
What happens when the base is zero?
Zero raised to any positive exponent equals zero: $0^n = 0$ for $n > 0$. Zero raised to a negative exponent is undefined (division by zero). The expression $0^0$ is mathematically indeterminate.
Is this calculator private?
Yes. All calculations run locally in your browser. No data is transmitted, logged, or stored on any server.