Chapter 7, Problem 22CR

To determine

To fill:The blank and complete the equation $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^{?}{n}^{?}=m^{\frac{5}{4}}n$ .

Answer to Problem 22CR

The equation is $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^{\frac{1}{2}}{n}^{\frac{1}{2}}=m^{\frac{5}{4}}n$ .

Explanation of Solution

Given:

Theequation is $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^{?}{n}^{?}=m^{\frac{5}{4}}n$ .

Calculation:

Consider the equation as $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^x{n}^y=m^{\frac{5}{4}}n$ .

Simplify the equation $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^x{n}^y=m^{\frac{5}{4}}n$ to obtain the value of x and yas follows:

  $\begin{array}{c}{m}^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^x{n}^y=m^{\frac{5}{4}}n\\ m^{\frac{3}{4}+x}{n}^{\frac{1}{2}+y}=m^{\frac{5}{4}}n\end{array}$

Compare the exponents of mon both sides of the equation.

  $\begin{array}{c}\frac{3}{4}+x=\frac{5}{4}\\ x=\frac{5}{4}-\frac{3}{4}\\ x=\frac{2}{4}\\ x=\frac{1}{2}\end{array}$

Compare the exponents of n on both sides of the equation.

  $\begin{array}{c}\frac{1}{2}+y=1\\ y=1-\frac{1}{2}\\ y=\frac{1}{2}\end{array}$

Therefore, the value of x is $\frac{1}{2}$ and y is $\frac{1}{2}$ .

Thus, the equation is $m^{\frac{3}{4}}{n}^{\frac{1}{2}}\cdot m^{\frac{1}{2}}{n}^{\frac{1}{2}}=m^{\frac{5}{4}}n$ .