Answer
The simplified form of the rational expression $\frac{5}{x+7}\cdot \frac{x+7}{10}$ is $\frac{1}{2}$.
Work Step by Step
Consider the provided rational expression,
$\frac{5}{x+7}\cdot \frac{x+7}{10}$
The quotient rule for two rational expressions is,
$\frac{A}{B}\div \frac{C}{D}=\frac{A}{B}\cdot \frac{D}{C}$
So, multiply the divisor,
$\frac{5}{x+7}\cdot \frac{x+7}{10}=\frac{5\left( x+7 \right)}{\left( x+7 \right)10}$
Regroup and remove the factor equal to 1,
$\begin{align}
& \frac{5}{x+7}\cdot \frac{x+7}{10}=\frac{\left( 5 \right)\left( x+7 \right)}{\left( x+7 \right)\left( 10 \right)} \\
& =\frac{\left( x+7 \right)\left( 5 \right)}{\left( x+7 \right)\left( 10 \right)} \\
& =1\cdot \frac{\left( 1 \right)}{\left( 2 \right)} \\
& =\frac{1}{2}
\end{align}$
