Exponent Rules Cheat Sheet: All 8 Rules with Examples

Whether you are simplifying algebraic expressions, solving equations, or working with scientific notation, exponent rules come up constantly. This cheat sheet puts every rule in one place with quick-reference formulas, worked examples, and common pitfalls.

Bookmark this page and come back whenever you need a refresher.

The 8 Essential Exponent Rules

All of the following assume $a, b \neq 0$ and exponents are real numbers unless otherwise noted.

Rule 1: Product Rule

a^m \cdot a^n = a^{m+n}

When multiplying like bases, add the exponents.

Quick example: $x^4 \cdot x^3 = x^{4+3} = x^7$

Rule 2: Quotient Rule

\frac{a^m}{a^n} = a^{m-n}

When dividing like bases, subtract the exponents.

Quick example: $\frac{y^9}{y^5} = y^{9-5} = y^4$

Rule 3: Power of a Power

(a^m)^n = a^{m \cdot n}

When raising a power to another power, multiply the exponents.

Quick example: $(2^3)^4 = 2^{12} = 4096$

Rule 4: Power of a Product

(ab)^n = a^n \cdot b^n

Distribute the exponent to each factor inside the brackets.

Quick example: $(3x)^2 = 3^2 \cdot x^2 = 9x^2$

Rule 5: Power of a Quotient

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Distribute the exponent to both numerator and denominator.

Quick example: $\left(\frac{2}{5}\right)^3 = \frac{8}{125}$

Rule 6: Zero Exponent

a^0 = 1 \quad (a \neq 0)

Any non-zero number raised to the power of zero equals 1.

Why: $\frac{a^n}{a^n} = a^{n-n} = a^0 = 1$

Rule 7: Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent means “take the reciprocal.”

Quick example: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$

Rule 8: Fractional Exponent

a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m

The denominator of the fractional exponent indicates the root, and the numerator indicates the power.

Quick example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$

Practice Problems with Solutions

Problem 1

Simplify $\frac{x^5 \cdot x^{-2}}{x^4}$ .

\frac{x^{5+(-2)}}{x^4} = \frac{x^3}{x^4} = x^{3-4} = x^{-1} = \frac{1}{x}

Problem 2

Simplify $\left(\frac{2a^3 b}{4a b^2}\right)^2$ .

First simplify inside the brackets:

\frac{2a^3 b}{4a b^2} = \frac{a^2}{2b}

Then apply the exponent:

\left(\frac{a^2}{2b}\right)^2 = \frac{a^4}{4b^2}

Problem 3

Evaluate $(16)^{3/4}$ .

16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8

Problem 4

Simplify $\frac{(3^2)^3 \cdot 3^{-4}}{3^5}$ .

\frac{3^6 \cdot 3^{-4}}{3^5} = \frac{3^{6-4}}{3^5} = \frac{3^2}{3^5} = 3^{-3} = \frac{1}{27}

Problem 5

Write $\sqrt{x} \cdot \sqrt[3]{x}$ as a single power of $x$ .

x^{1/2} \cdot x^{1/3} = x^{1/2 + 1/3} = x^{3/6 + 2/6} = x^{5/6}

The 5 Most Common Mistakes

Adding exponents with different bases. $2^3 \cdot 3^2 \neq 6^5$ . The product rule only works with the same base.
Distributing exponents over addition. $(a + b)^2 \neq a^2 + b^2$ . You must expand using FOIL or the binomial theorem.
Confusing $-x^2$ with $(-x)^2$ . The first is $-(x^2)$ (always negative), while the second is $x^2$ (always positive).
Saying $0^0 = 0$ . The expression $0^0$ is indeterminate in most mathematical contexts, though it is conventionally defined as 1 in combinatorics and series.
Forgetting the fractional exponent rule. $x^{1/2}$ is $\sqrt{x}$ , not $x/2$ .

Quick-Reference Summary Table

Here is every rule at a glance:

Rule	Formula
Product	$a^m \cdot a^n = a^{m+n}$
Quotient	$a^m / a^n = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$
Power of a product	$(ab)^n = a^n b^n$
Power of a quotient	$(a/b)^n = a^n / b^n$
Zero exponent	$a^0 = 1$
Negative exponent	$a^{-n} = 1/a^n$
Fractional exponent	$a^{m/n} = \sqrt[n]{a^m}$

Try It Yourself

Evaluate any exponent expression with our free Exponent Calculator. It handles integer, fractional, and negative exponents and shows step-by-step working. For logarithmic expressions (the inverse of exponents), try our Log Calculator.