Question

Evaluate $\frac{2a^{3}b}{a^2+2ab+b^2}$for each value:

$ⓐa=-1,b=2ⓑa=0,b=-1ⓒa=1,b=\frac{1}{2}$

Short answer

$$\begin{gathered} ⓐ\mathrm{The}\mathrm{value}\mathrm{of}\frac{2a^{3}b}{a^2+2ab+b^2}\mathrm{for}a=-1,b=2\mathrm{is}-4. \\ ⓑ\mathrm{The}\mathrm{value}\mathrm{of}\frac{2a^{3}b}{a^2+2ab+b^2}\mathrm{for}a=0,b=-1\mathrm{is}0. \\ ⓒ\mathrm{The}\mathrm{value}\mathrm{of}\frac{2a^{3}b}{a^2+2ab+b^2}\mathrm{for}a=1,b=\frac{1}{2}\mathrm{is}\frac{4}{9}. \end{gathered}$$

Step-by-step solution

Step 1. Given Information 

Evaluate $\frac{2a^{3}b}{a^2+2ab+b^2}$for each value:

$ⓐa=-1,b=2ⓑa=0,b=-1ⓒa=1,b=\frac{1}{2}$

Part (a) Step 1. Finding the value of 2a3ba2+2ab+b2 for a=−1, b=2

Putting $a=-1,b=2$on the given expression

role="math" localid="1648380773911" $$\begin{gathered} \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{2·{(-1)}^3·2}{{(-1)}^2+2·(-1)·2+{\left(2\right)}^2} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{-4}{1-4+4} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{-4}{1} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=-4 \end{gathered}$$

Part (b) Step 1. Finding the value of 2a3ba2+2ab+b2 for a=0, b=−1

Putting $a=0,b=-1$on the given expression

role="math" localid="1648380909116" $$\begin{gathered} \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{2{·\left(0\right)}^3·(-1)}{{\left(0\right)}^2+2·0·(-1)+{(-1)}^2} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{0}{0-0+1} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{0}{1} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=0 \end{gathered}$$

Part (c) Step 1. Finding the value of 2a3ba2+2ab+b2 for a=1, b=12

Putting $a=1,b=\frac{1}{2}$on the given expression

$$\begin{gathered} \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{2·{\left(1\right)}^3·\frac{1}{2}}{{\left(1\right)}^2+2·\left(1\right)·\frac{1}{2}+{\left(\frac{1}{2}\right)}^2} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{1}{1+1+\frac{1}{4}} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{1}{1·\frac{4}{4}+1·\frac{4}{4}+\frac{1}{4}} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{1}{\frac{4}{4}+\frac{4}{4}+\frac{1}{4}} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{1}{\frac{4+4+1}{4}} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{1}{\frac{9}{4}} \\ \frac{2a^{3}b}{a^2+2ab+b^2}=\frac{4}{9} \end{gathered}$$