Chapter 7, Problem 21CR

To determine

To fill:The blank and complete the equation $x^{\frac{2}{3}}\cdot x^{?}=x^{\frac{11}{12}}$ .

Answer to Problem 21CR

The equation is $x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}=x^{\frac{11}{12}}$ .

Explanation of Solution

Given:

Theequation is $x^{\frac{2}{3}}\cdot x^{?}=x^{\frac{11}{12}}$ .

Calculation:

Consider the equation as $x^{\frac{2}{3}}\cdot x^p=x^{\frac{11}{12}}$ .

Simplify the equation $x^{\frac{2}{3}}\cdot x^p=x^{\frac{11}{12}}$ to obtain the value of pas follows:

  $\begin{array}{c}{x}^{\frac{2}{3}}\cdot x^p=x^{\frac{11}{12}}\\ x^{\frac{2}{3}+p}=x^{\frac{11}{12}}\end{array}$

Compare the exponents on both sides of the equation.

  $\begin{array}{c}\frac{2}{3}+p=\frac{11}{12}\\ p=\frac{11}{12}-\frac{2}{3}\\ p=\frac{11-8}{12}\\ p=\frac{3}{12}\\ p=\frac{1}{4}\end{array}$

Therefore, the value of p is $\frac{1}{4}$ .

Thus, the equation is $x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}=x^{\frac{11}{12}}$ .