Exponents Calculator

What an exponent means

An exponent (or power) tells you how many times to multiply a number by itself. In $x^n$, $x$ is the base and $n$ is the exponent:

$x^n = \underbrace{x \cdot x \cdots x}_{n \text{ times}}$

For example, $3^4 = 3 \times 3 \times 3 \times 3 = 81$ — read as "3 to the power of 4."

The sign trap: $-4^2$ vs. $(-4)^2$

This is one of the most common mistakes in algebra, because the two expressions look almost identical but mean different things:

Without parentheses, the exponent applies only to the $4$, and the minus sign is applied after squaring — so $-4^2 = -(4^2) = -16$. With parentheses, the minus sign is part of the base being squared, so $(-4)^2 = (-4) \times (-4) = 16$.

This calculator always reads a negative number typed into the base field as the whole base. Entering $-4$ with exponent $2$ computes $(-4)^2 = 16$, the same way a graphing calculator treats a parenthesized negative base.

Negative exponents mean reciprocals

A negative exponent does not make the result negative — it means "take the reciprocal":

$x^{-m} = \frac{1}{x^m}$

So $3^{-4} = \frac{1}{3^4} = \frac{1}{81} \approx 0.012346$ — a small positive fraction, not $-81$. This calculator shows both the exact reduced fraction and its decimal value for negative integer exponents.

Zero and one as exponents

• $x^1 = x$ — any number to the power 1 is itself.
• $x^0 = 1$ for any nonzero $x$ — this holds no matter how large or small $x$ is.
• $0^0 = 1$ by convention, though it is sometimes left undefined in more advanced contexts.

Decimal bases and fractional exponents

The base does not need to be a whole number: $2.5^2 = 2.5 \times 2.5 = 6.25$, computed exactly.

A fractional exponent represents a root: $x^{1/n} = \sqrt[n]{x}$, and more generally $x^{m/n} = \sqrt[n]{x^m}$. For example, $4^{0.5} = \sqrt{4} = 2$. This calculator gives the decimal approximation for fractional exponents — for the exact radical form (like $\sqrt{2}$ instead of $1.414213562$), use the Fractional Exponents calculator.

A negative base raised to a non-integer exponent — like $-4$ to the power $0.5$ — has no real-number result, since it is equivalent to an even root of a negative number. This calculator reports that combination as invalid rather than guessing.

The seven exponent rules

These rules apply to every base and exponent this calculator can compute:

Worked examples

Example 1 — positive integer exponent: $3^4 = 3 \times 3 \times 3 \times 3 = 81$.

Example 2 — negative integer exponent: $3^{-4} = \frac{1}{3^4} = \frac{1}{81} \approx 0.012346$.

Example 3 — the sign trap: $(-4)^2 = 16$ (square of $-4$), while $-4^2 = -16$ (negative of $4^2$) — same digits, opposite sign, because the parentheses change what gets squared.

Example 4 — decimal base: $2.5^2 = 6.25$.

Where exponents show up

Exponents drive compound growth (interest that compounds over time), area and volume formulas (squaring and cubing side lengths), and scientific notation, where very large or small numbers are written as $a \times 10^n$ — see the scientific notation converter for that specific case. Square (exponent 2) and cube (exponent 3) are common enough to get their own dedicated calculators, and very large results belong on the Large Exponents calculator, which keeps every digit of a big-integer power exact.