Practice Exponent Rules in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

A set of laws governing how exponents behave under multiplication, division, and raising to a power: product rule (amโ‹…an=am+na^m \cdot a^n = a^{m+n}), quotient rule (am/an=amโˆ’na^m / a^n = a^{m-n}), power rule ((am)n=amn(a^m)^n = a^{mn}), zero exponent (a0=1a^0 = 1 for aโ‰ 0a \neq 0), and negative exponent (aโˆ’n=1ana^{-n} = \frac{1}{a^n}).

Since a3=aโ‹…aโ‹…aa^3 = a \cdot a \cdot a and a2=aโ‹…aa^2 = a \cdot a, multiplying them gives aโ‹…aโ‹…aโ‹…aโ‹…a=a5a \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.

Showing a random 20 of 50 problems.

Example 1

medium
Simplify x5โ‹…x3x2\frac{x^5\cdot x^3}{x^2}.

Example 2

medium
Simplify (2x3)2โ‹…x(2x^3)^2\cdot x.

Example 3

easy
Simplify x5โ‹…x6x^5 \cdot x^6.

Example 4

hard
Justify the zero-exponent rule using the quotient rule: show a0=1a^0 = 1 for aโ‰ 0a \ne 0.

Example 5

medium
Simplify a3b2ab5\frac{a^3 b^2}{a b^5}.

Example 6

medium
Simplify 8x7y32x3y\frac{8x^7 y^3}{2x^3 y}.

Example 7

easy
Simplify x5โ‹…x3x2\frac{x^5 \cdot x^3}{x^2}.

Example 8

challenge
Simplify (xa)bโ‹…xcxab\frac{(x^a)^b \cdot x^c}{x^{ab}} assuming all variables represent positive integers.

Example 9

easy
Simplify y10y4\frac{y^{10}}{y^4}.

Example 10

challenge
Solve for xx: 32x+1=273^{2x+1} = 27.

Example 11

medium
Simplify (xโˆ’3)โˆ’2(x^{-3})^{-2}.

Example 12

medium
Simplify 24โ‹…2โˆ’62^4\cdot2^{-6}.

Example 13

easy
Write xโˆ’4x^{-4} with a positive exponent.

Example 14

medium
Simplify (x3x5)2\left(\frac{x^3}{x^5}\right)^2.

Example 15

easy
Evaluate 707^0.

Example 16

easy
Simplify 23โ‹…222^3\cdot2^2.

Example 17

hard
Solve for nn: 2nโ‹…25=2122^n \cdot 2^5 = 2^{12}.

Example 18

easy
Simplify (a2)3(a^2)^3.

Example 19

medium
Simplify (a2b)3\left(\frac{a^2}{b}\right)^3.

Example 20

medium
Simplify (2x3y)4(2x^3y)^4.