Beginning Algebra With Applications, Chapter 7, 7.2, Section 7.2, Problem 90

Simplify $\displaystyle 5a^2 b (ab^2)^2 + b^3 (2a^2 b)^2$


$
\begin{equation}
\begin{aligned}

5a^2 b (ab^2)^2 + b^3 (2a^2 b)^2 =& 5a^2 b \left( a^2 b^4 \right) + b^3 (2)^2 \left( a^4 b^2 \right)
&& \text{Multiply each exponent in $ab^2$ and in $2a^2 b$ by the exponent outside the parentheses}
\\
\\
=& 5a^2 b \left( a^2 b^4 \right) + b^3 \left( 4a^4 b^2 \right)
&& \text{Simplify } (2)^2
\\
\\
=& 5 \left( a^2 \cdot a^2 \right) (b \cdot b^4) + 4 (a^4) \left( b^3 \cdot b^2 \right)
&& \text{Use Properties of Multiplication to rearrange and group factors}
\\
\\
=& 5a^4 b^5 + 4a^4 b^5
&& \text{Multiply variables with the same base by adding the exponents}
\\
\\
=& 9a^4 b^5
&&

\end{aligned}
\end{equation}
$