Chapter 4.4, Problem 1MRE

To determine

To simplify: the expression

  $\left(3a^{2}b\right)\left(2ab\right)\left(4b\right)$

Answer to Problem 1MRE

  $24a^3{b}^3$

Explanation of Solution

Given:

  $\left(3a^{2}b\right)\left(2ab\right)\left(4b\right)$

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 : ${\left(ab\right)}^n=a^n{b}^n$

  If an expression contains more than one operations between its terms and brackets, first solve the terms inside brackets and then apply division followed by multiplication, addition and subtraction.

Calculation:

To simplify the expression

  $\left(3a^{2}b\right)\left(2ab\right)\left(4b\right)$

Use the basic properties of power.

First simplify this using the property $a^n\cdot a^m=a^{m+n}$ , and then proceed further to simplify, as

  $\begin{array}{c}\left(3a^{2}b\right)\left(2ab\right)\left(4b\right)=3b\cdot 2a^{2+1}b\cdot 4b\\ =3b\cdot 2a^{3}b\cdot 4b\\ =3\cdot 2a^3\cdot 4b^3;\text{ using property }{a}^m{a}^n=a^{m+n}\\ =24a^3{b}^3\end{array}$

Thus, simplified form of given expression is

  $\left(3a^{2}b\right)\left(2ab\right)\left(4b\right)=24a^3{b}^3$