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.

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: $$ \boxed{ \frac{ x^a } { x^ b }= x^{a \red - b} } \\ \text{Example : } \\ \frac{ 7^6 } { 7^ 2 }= 7^{6 \red - 2} \\ = 7^4 $$

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

seperate lesson### 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

Since we are multiplying exponents we add the exponents:

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

##### Problem 2

Use the power rule of exponents and multiply exponents:

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

##### Problem 3

Use the quotient rule of exponents and subtract the exponents:

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

##### Problem 4

Since we are multiplying exponents we add the exponents

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

##### Problem 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

#### Negative Exponents

#### Fraction Exponents

Fraction exponents are explained separately here