Chapter 5.2, Problem 39WE

To determine

To find: the missing factor ${\left(3a^3{b}^2\right)}^3={\left(3a^3{b}^2\right)}^2\left(\_\_\_\_\right)$

Answer to Problem 39WE

  $3a^3{b}^2$

Explanation of Solution

Given:

The expression: ${\left(3a^3{b}^2\right)}^3={\left(3a^3{b}^2\right)}^2\left(\_\_\_\_\right)$

Formula Used:

1. Property of quotients:

For real numbers a , b , c and d , such that $b\ne 0$ and $d\ne 0$ ,

  $\frac{ac}{bd}=\frac{a}{b}\cdot \frac{c}{d}$

2. For non zero real number a and positive integers m and n , such that $n>m$

  $\frac{a^n}{a^m}=a^{n-m}$

Calculation:

The required missing factor is found by dividing ${\left(3a^3{b}^2\right)}^3$ by ${\left(3a^3{b}^2\right)}^2$ .

That is, the missing factor is: $\frac{{\left(3a^3{b}^2\right)}^3}{{\left(3a^3{b}^2\right)}^2}$

Now use the Property of quotients and the formula $\frac{a^n}{a^m}=a^{n-m}$ , to simplify $\frac{{\left(3a^3{b}^2\right)}^3}{{\left(3a^3{b}^2\right)}^2}$ .

Consider,

  $\begin{array}{c}\frac{{\left(3a^3{b}^2\right)}^3}{{\left(3a^3{b}^2\right)}^2}={\left(3a^3{b}^2\right)}^{3-2}\\ ={\left(3a^3{b}^2\right)}^1\\ =3a^3{b}^2\end{array}$