# Laws of Exponents

How to simplify expressions using the laws of exponents

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

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

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

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### 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!