Chapter 4.4, Problem 32WE

To determine

To simplify: the expression

  ${\left(5a^{2}b\right)}^2{\left(5b\right)}^3$

Answer to Problem 32WE

  $3125a^6{b}^5$

Explanation of Solution

Given:

  ${\left(5a^{2}b\right)}^2{\left(5b\right)}^3$

Concept Used:

• For all positive integers m and n: ${\left(a^m\right)}^n=a^{mn}$
• For all integers m and n : $a^m{a}^n=a^{m+n}$
• For all integers m and n : ${\left(ab\right)}^n=a^n{b}^n$

Calculation:

To simplify the expression

  ${\left(5a^{2}b\right)}^2{\left(5b\right)}^3$

Use the basic properties of power.

First simplify this using the property ${\left(ab\right)}^n=a^n{b}^n$ , and then proceed further to simplify, as

  $\begin{array}{c}{\left(5a^{2}b\right)}^2{\left(5b\right)}^3=5^2{\left(a^2\right)}^3{b}^2\cdot 5^3{b}^3\\ =25a^{2\times 3}{b}^{2+3}\cdot 125;\text{ using properties }\left(\begin{array}{l}{\left(a^m\right)}^n=a^{mn},\\ \text{and }{a}^m{a}^n=a^{m+n}\end{array}\right)\\ =3125a^6{b}^5\end{array}$

Thus, simplified form of given expression is

  ${\left(5a^{2}b\right)}^2{\left(5b\right)}^3=3125a^6{b}^5$