Question

Rewrite as equivalent rational expressions with denominator

$\left(x+3\right)\left(x+5\right)\left(x+6\right)$

$\frac{x}{x^2+8x+15},\frac{5}{x^2+9x+18}$.

Short answer

The simplification of the expression is$\frac{x^2+6x}{\left(x+3\right)\left(x+5\right)\left(x+6\right)},\frac{5x+25}{\left(x+3\right)\left(x+5\right)\left(x+6\right)}$

Step-by-step solution

Given information 

The given rational expression is$\frac{x}{x^2+8x+15},\frac{5}{x^2+9x+18}$.

The simplification of the expression 

$\frac{x}{x^2+5x+3x+15},\frac{5}{x^2+9x+18}$

Factorize the denominator

role="math" localid="1648482456775" $$\begin{gathered} \frac{x}{x^2+5x+3x+15},\frac{5}{x^2+6x+3x+18} \\ \frac{x}{x\left(x+5\right)+3\left(x+5\right)},\frac{5}{x\left(x+6\right)+3\left(x+6\right)} \\ \frac{x}{\left(x+3\right)\left(x+5\right)},\frac{5}{\left(x+3\right)\left(x+6\right)} \end{gathered}$$

The LCD of the denominator $\left(x+3\right)\left(x+5\right)\left(x+6\right)$.

$$\begin{gathered} \frac{x\left(x+6\right)}{\left(x+3\right)\left(x+5\right)\left(x+6\right)},\frac{5\left(x+5\right)}{\left(x+3\right)\left(x+5\right)\left(x+6\right)} \\ Simplifythenumerator \\ \frac{x^2+6x}{\left(x+3\right)\left(x+5\right)\left(x+6\right)},\frac{5x+25}{\left(x+3\right)\left(x+5\right)\left(x+6\right)} \end{gathered}$$