Chapter 2.1, Problem 45E

Solution

To find: The number to fill in the blank $\left(?\right)$ of an expression.

The missing number for fill the blank is $\left(a^{-4}\cdot b^9\right)$ .

Given data: The given expression is ${\left(a^5{b}^4\right)}^2=a^{14}{b}^{-1}\cdot ?$ .

Formula used:

Laws of Indices: For numbers a , b , m and n the following statements are true:

  $\begin{array}{c}{\left(a^m{b}^n\right)}^k=a^{mk}{b}^{nk}\\ \left(a^p{a}^q\right)\left(b^r{b}^s\right)=\left(a^{p+q}\right)\left(b^{r+s}\right)\end{array}$

Calculation:

The given statement is ${\left(a^5{b}^4\right)}^2=a^{14}{b}^{-1}\cdot ?$ .

Consider left hand side of the statement which using laws of indices can be written as:

  $\begin{array}{c}{\left(a^5\cdot b^4\right)}^2=a^{5\times 2}\cdot b^{4\times 2}\\ =a^{10}\cdot b^8\\ =a^{14-4}\cdot b^{-1+9}\\ =a^{14}\cdot a^{-4}\cdot b^{-1}\cdot b^9\\ =\left(a^{14}\cdot b^{-1}\right)\cdot \left(a^{-4}\cdot b^9\right)\end{array}$

Therefore, the missing number is $\left(a^{-4}\cdot b^9\right)$ .