Exponent Calculator (xⁿ) – Solve Powers, Roots & Indices

Exponent Calculator

Quickly calculate powers and see the step-by-step breakdown.

Base Number Use Euler’s number (e)

Exponent (Power)

The Ultimate Guide to Exponents

What is an Exponent?

At its core, an exponent is a mathematical shorthand for repeated multiplication. When you see a number written with a smaller number raised to its upper right, you are looking at an exponent in action.

The large number at the bottom is the base (the number being multiplied), and the small number at the top is the exponent or power (how many times the base is multiplied by itself).

For example, if you have $5^3$, it simply means:

$$5^3 = 5 \times 5 \times 5 = 125$$

Essential Exponent Rules & Laws

Mathematics has a few handy shortcuts when dealing with exponents. Here are the core rules you need to know, along with practical examples.

1. The Product Rule (Multiplying)

When you multiply two expressions that have the same base, you simply add their exponents together.

$$x^a \times x^b = x^{a+b}$$

Example: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$

2. The Quotient Rule (Dividing)

When you divide two expressions with the same base, you subtract the bottom exponent from the top exponent.

$$\frac{x^a}{x^b} = x^{a-b}$$

Example: $\frac{4^5}{4^3} = 4^{5-3} = 4^2 = 16$

3. Power of a Power Rule

If you have a base raised to an exponent, and that entire expression is raised to another exponent, you multiply the exponents.

$$(x^a)^b = x^{a \times b}$$

Example: $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$

4. The Zero Exponent Rule

This is a favorite among mathematicians: any non-zero number raised to the power of 0 is always equal to 1.

$$x^0 = 1$$

Example: $999^0 = 1$

5. Negative Exponent Rule

A negative exponent tells you to flip the base to the other side of a fraction line (take the reciprocal) and make the exponent positive.

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

Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

6. Fractional Exponents

When an exponent is a fraction, it represents a root. The denominator tells you which root to take, and the numerator acts as a standard power.

$$x^{\frac{a}{b}} = \sqrt[b]{x^a}$$

Example: $8^{\frac{1}{3}} = \sqrt[3]{8^1} = 2$