Question

Simplify each expression.

$\frac{4y^2}{8y^4+16y^3}$

Short answer

The simplified expression is$\frac{1}{2y^2+4y}$.

Step-by-step solution

Step 1. Define rational expression.

A rational expression is an algebraic expression whose numerator and denominator are polynomials. Also, division by 0 is undefined, so therefore, the polynomials in the denominator cannot be zero and the values for which the denominator is zero is termed as excluded values for the respective rational expression.

Step 2. Factor the expression.

Factor the numerator and denominator, using their greatest common factor in the expression $\frac{4y^2}{8y^4+16y^3}$.

role="math" localid="1648041783563" $$\begin{gathered} \frac{4y^2}{8y^4+16y^3}=\frac{4y^2}{8y^3\left(y+2\right)} \\ =\frac{4y^2}{4y^2\left(2y\right)\left(y+2\right)} \end{gathered}$$

Step 3. Simplify the expression.

Further simplify the expression.

$$\begin{gathered} \frac{4y^2}{8y^4+16y^3}=\frac{4y^2}{4y^2\left(2y\right)\left(y+2\right)} \\ =\frac{1}{\left(2y\right)\left(y+2\right)} \\ =\frac{1}{2y^2+4y} \end{gathered}$$

Therefore, the simplified expression is $\frac{1}{2y^2+4y}$.