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Laws of Exponents

How to simplify expressions using the laws of exponents

There are many different laws of exponents. This page covers the 3 most frequently studied formulas in Algebra I. If you are looking for other laws, visit our exponents home page.

Video on the Laws of Exponents

There are many different laws of exponents. This page covers the 3 most frequently studied laws of exponents (Rules 1-3 below).

Rule 1: $$ \boxed{ x^a \cdot x^ b = x^{a \red + b} } \\ \text{Example : } \\ 3^4 \cdot 3^2 = 3^{4+2} \\ 3^4 \cdot 3^2 = 3^{6} $$


Rule 2: $$ \boxed{ \frac{ x^a } { x^ b }= x^{a \red - b} } \\ \text{Example : } \\ \frac{ 7^6 } { 7^ 2 }= 7^{6 \red - 2} \\ = 7^5 $$


Rule 3: $$ \boxed{ (x^a)^ b = x^{a \red \cdot b} } \\ \text{Example : } \\ (3^2)^ 4 = 3^{2 \red \cdot 4} \\ = 3^{8} $$


Rule 4: $$ \boxed{ x^{- \red a} = \frac {1}{x^{\red a}} } $$

seperate lesson

Rule 5: $$ \boxed{ x^{\frac m n} = ? } $$

seperate lesson fractional Exponent

Be Careful

be Careful

Every rule above only works when the bases are the same.

So, something like $$ \red 3 ^2 \cdot \red 5^3 $$ can not be simplified using these rules because the bases are different

Practice Problems

Problem 1

Simplify $ x^3 \cdot x^2 $

Since we are multiplying exponents we add the exponents:

$ x^3 \cdot x^2 = x^{3+2} = x^5 $

Problem 2

Simplify $ (x^3) ^2 $

Use the power rule of exponents and multiply exponents:

$ (x^3) ^2 = x^{(3 \cdot 2)}= x^6 $

Problem 3

Evaluate $ \frac {x^5}{ x^3} $

Use the quotient rule of exponents and subtract the exponents:

$ \frac {x^5}{ x^3} = x^{(5 - 3)}= x^2 $

Problem 4

Simplify $ x^{11} \cdot x^5 $

Since we are multiplying exponents we add the exponents

$ x^{11} \cdot x^5 = x^{11+5} = x^{16} $

Problem 5

Simplify $ x^8 \cdot y^5 $

You cannot simplify this expression! Remember that all of the rules on this page only work when the exponents have the same base!

Exponent Rules Covered Elsewhere


Back to Exponents Next to Subtracting Exponents