Exponent Rules Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exponent Rules.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A set of laws governing how exponents behave under multiplication, division, and raising to a power: product rule ( $a^m \cdot a^n = a^{m+n}$ ), quotient rule ( $a^m / a^n = a^{m-n}$ ), power rule ( $(a^m)^n = a^{mn}$ ), zero exponent ( $a^0 = 1$ for $a \neq 0$ ), and negative exponent ( $a^{-n} = \frac{1}{a^n}$ ).

Since $a^3 = a \cdot a \cdot a$ and $a^2 = a \cdot a$ , multiplying them gives $a \cdot a \cdot a \cdot a \cdot a = a^5$ . You just add the counts. All the other rules follow the same logic of counting how many times you multiply.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every exponent rule is bookkeeping for repeated factors.

Common stuck point: The procedure for exponent rules is the easy part; the trap is adding exponents when terms are added. Asking "What operation is happening to the powers?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: What operation is happening to the powers?

Worked Examples

Example 1

easy

Simplify

\frac{x^5 \cdot x^3}{x^2}

Answer

x^6

First step

Combine the factors in the numerator with the product rule:

x^5 \cdot x^3 = x^{5+3} = x^8

Full solution

2
Rewrite the expression as $\frac{x^8}{x^2}$ and apply the quotient rule for like bases.
3
Subtract the exponents: $x^{8-2} = x^6$ .

The product rule says

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

, and the quotient rule says

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

. These two laws handle most exponent simplifications.

Example 2

medium

Simplify

(2x^3y)^4

Example 3

easy

Simplify

2^4 \cdot 2^3

Example 4

medium

Simplify

\frac{6x^5 y^{-2}}{3x^2 y^3}

using only positive exponents.

Example 5

hard

Justify the zero-exponent rule using the quotient rule: show

a^0 = 1

for

a \ne 0

Example 6

hard

Show that

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

by writing out factors when

a = 5, m = 3, n = 2

Example 7

challenge

Solve for

x

3^{2x+1} = 27

Practice Problems

5^0

Example 7

easy

Write

a^{-3}

2^3\cdot3^3

NOT equal to

6^6

? Simplify it correctly.

Example 16

medium

Simplify

(2x^3)^2\cdot x

Example 17

medium

Evaluate

\frac{3^5}{3^5}

two ways and reconcile.

Example 18

challenge

Solve for

n

2^n=\frac{1}{8}

Example 19

challenge

Simplify

\frac{a^{2n+1}\cdot a^{n-1}}{a^{3n}}

in terms of

n

Example 20

challenge

a^x=3

and

a^y=5

, find

a^{2x+y}

Example 21

medium

Simplify

\frac{a^3 b^2}{a b^5}

Example 22

medium

Simplify

(a^2 b)^3 \cdot a

Example 23

easy

Simplify

x^5 \cdot x^6

Example 24

easy

Simplify

\frac{y^{10}}{y^4}

Example 25

easy

Write

x^{-4}

n

2^n \cdot 2^5 = 2^{12}

Example 36

hard

Express

\frac{1}{16}

as a power of 2 with a negative exponent.

Example 37

hard

Simplify

(2x^{-2}y)^{-3}

with positive exponents only.

Example 38

challenge

Simplify

\frac{(x^a)^b \cdot x^c}{x^{ab}}

assuming all variables represent positive integers.

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Related Concepts

Exponents Multiplication Division Scientific Notation Exponential Function Polynomials

Background Knowledge

These ideas may be useful before you work through the harder examples.

exponentsmultiplicationdivision