Comme un jeu

Simplifying Powers of Products

Exponent Law 4

Let's look at an example:$$\begin{aligned}(\textcolor{colordef}{3} \times \textcolor{colorprop}{5})^{\textcolor{olive}{2}}&= (\textcolor{colordef}{3} \times \textcolor{colorprop}{5}) \times (\textcolor{colordef}{3} \times \textcolor{colorprop}{5}) \\ &= \textcolor{colordef}{3} \times \textcolor{colorprop}{5} \times \textcolor{colordef}{3} \times \textcolor{colorprop}{5} \\ &= (\textcolor{colordef}{3} \times \textcolor{colordef}{3}) \times (\textcolor{colorprop}{5} \times \textcolor{colorprop}{5}) \\ &= \textcolor{colordef}{3}^{\textcolor{olive}{2}}\, \textcolor{colorprop}{5}^{\textcolor{olive}{2}}\end{aligned}$$In general, when you multiply two numbers $\textcolor{colordef}{a}$ and $\textcolor{colorprop}{b}$, and then raise the product to the power $\textcolor{olive}{n}$, that is$$(\textcolor{colordef}{a}\textcolor{colorprop}{b})^{\textcolor{olive}{n}},$$the result is each factor raised to the power $\textcolor{olive}{n}$:$$(\textcolor{colordef}{a}\textcolor{colorprop}{b})^{\textcolor{olive}{n}} = \textcolor{colordef}{a}^{\textcolor{olive}{n}}\, \textcolor{colorprop}{b}^{\textcolor{olive}{n}}.$$

Proposition Exponent Law 4

For any numbers $n$ and any numbers $a$ and $b$,$$\left(\textcolor{colordef}{a}\textcolor{colorprop}{b}\right)^{\textcolor{olive}{n}} = \textcolor{colordef}{a}^{\textcolor{olive}{n}}\, \textcolor{colorprop}{b}^{\textcolor{olive}{n}}$$

Example

Simplify $(\textcolor{colordef}{2}\times \textcolor{colorprop}{5})^{\textcolor{olive}{3}}$.

Answer

$$(\textcolor{colordef}{2}\times \textcolor{colorprop}{5})^{\textcolor{olive}{3}}= \textcolor{colordef}{2}^{\textcolor{olive}{3}}\, \textcolor{colorprop}{5}^{\textcolor{olive}{3}}$$

Exponent Law 4

Exercise

Simplify:$(2\times x)^{3} =$

Answer

$$\begin{aligned}(\textcolor{colordef}{2}\times \textcolor{colorprop}{x})^{\textcolor{olive}{3}}&= (\textcolor{colordef}{2}\times \textcolor{colorprop}{x}) \times (\textcolor{colordef}{2}\times \textcolor{colorprop}{x}) \times (\textcolor{colordef}{2}\times \textcolor{colorprop}{x}) \\ &= (\textcolor{colordef}{2} \times \textcolor{colordef}{2} \times \textcolor{colordef}{2}) \times (\textcolor{colorprop}{x} \times \textcolor{colorprop}{x} \times \textcolor{colorprop}{x}) \\ &= \textcolor{colordef}{2}^{\textcolor{olive}{3}} \times \textcolor{colorprop}{x}^{\textcolor{olive}{3}} \\ \end{aligned}$$

Exercise

Simplify:$(x\times 3)^{2} =$

Answer

$$\begin{aligned}(\textcolor{colordef}{x}\times \textcolor{colorprop}{3})^{\textcolor{olive}{2}}&= (\textcolor{colordef}{x}\times \textcolor{colorprop}{3}) \times (\textcolor{colordef}{x}\times \textcolor{colorprop}{3}) \\ &= (\textcolor{colordef}{x} \times \textcolor{colordef}{x}) \times (\textcolor{colorprop}{3} \times \textcolor{colorprop}{3}) \\ &= \textcolor{colordef}{x}^{\textcolor{olive}{2}} \times \textcolor{colorprop}{3}^{\textcolor{olive}{2}} \\ \end{aligned}$$

Exercise

Simplify:$(5\times x)^{4} =$

Answer

$$\begin{aligned}(\textcolor{colordef}{5}\times \textcolor{colorprop}{x})^{\textcolor{olive}{4}}&= (\textcolor{colordef}{5}\times \textcolor{colorprop}{x}) \times (\textcolor{colordef}{5}\times \textcolor{colorprop}{x}) \times (\textcolor{colordef}{5}\times \textcolor{colorprop}{x}) \times (\textcolor{colordef}{5}\times \textcolor{colorprop}{x}) \\ &= (\textcolor{colordef}{5} \times \textcolor{colordef}{5} \times \textcolor{colordef}{5} \times \textcolor{colordef}{5}) \times (\textcolor{colorprop}{x} \times \textcolor{colorprop}{x} \times \textcolor{colorprop}{x} \times \textcolor{colorprop}{x}) \\ &= \textcolor{colordef}{5}^{\textcolor{olive}{4}} \times \textcolor{colorprop}{x}^{\textcolor{olive}{4}} \\ \end{aligned}$$

Exercise

Simplify:$(x\times 2)^{5} =$

Answer

$$\begin{aligned}(\textcolor{colordef}{x}\times \textcolor{colorprop}{2})^{\textcolor{olive}{5}}&= (\textcolor{colordef}{x}\times \textcolor{colorprop}{2}) \times (\textcolor{colordef}{x}\times \textcolor{colorprop}{2}) \times (\textcolor{colordef}{x}\times \textcolor{colorprop}{2}) \times (\textcolor{colordef}{x}\times \textcolor{colorprop}{2}) \times (\textcolor{colordef}{x}\times \textcolor{colorprop}{2}) \\ &= (\textcolor{colordef}{x} \times \textcolor{colordef}{x} \times \textcolor{colordef}{x} \times \textcolor{colordef}{x} \times \textcolor{colordef}{x}) \times (\textcolor{colorprop}{2} \times \textcolor{colorprop}{2} \times \textcolor{colorprop}{2} \times \textcolor{colorprop}{2} \times \textcolor{colorprop}{2}) \\ &= \textcolor{colordef}{x}^{\textcolor{olive}{5}} \times \textcolor{colorprop}{2}^{\textcolor{olive}{5}} \\ \end{aligned}$$