Chapter 3.6, Problem 8Q

Solution

To calculate: The value of x for the equation $\frac{1}{4}{\left(7x+8\right)}^{\frac{3}{2}}=54$ and also check the solution.

The required answer is $x=4$ .

Given information:

The equation is $\frac{1}{4}{\left(7x+8\right)}^{\frac{3}{2}}=54$ .

Formula used:

Raise each side of the equation to the same power.

Calculation:

Write the original equation:

  $\frac{1}{4}{\left(7x+8\right)}^{\frac{3}{2}}=54$

Multiply each side of the equation with 4,

  $\begin{array}{c}\left(\frac{1}{4}{\left(7x+8\right)}^{\frac{3}{2}}\right)\cdot 4=54\cdot 4\\ {\left(7x+8\right)}^{\frac{3}{2}}=216\end{array}$

Raise each side to the power $\frac{2}{3}$ and simplify,

  $\begin{array}{c}{\left[{\left(7x+8\right)}^{\frac{3}{2}}\right]}^{\frac{2}{3}}={216}^{\frac{2}{3}}\\ 7x+8={216}^{\frac{2}{3}}\end{array}$

Rewrite using the property of exponent,

  $\begin{array}{c}7x+8={\left(6^3\right)}^{\frac{2}{3}}\\ 7x+8=6^2\\ 7x+8=36\end{array}$

Subtract 8 from both sides of the equation,

  $\begin{array}{c}7x=36-8\\ 7x=28\end{array}$

Divide each side by 7,

  $\begin{array}{c}\frac{7x}{7}=\frac{28}{7}\\ x=4\end{array}$

Now check $x=4$ in the original equation.

  $\frac{1}{4}{\left(7x+8\right)}^{\frac{3}{2}}=54$

Now substitute 4 for x and simplify,

  $\begin{array}{r}\frac{1}{4}{\left(7\cdot 4+8\right)}^{\frac{3}{2}}\stackrel{}{=}54\\ \frac{1}{4}{\left(28+8\right)}^{\frac{3}{2}}\stackrel{}{=}54\\ \frac{1}{4}{\left(36\right)}^{\frac{3}{2}}\stackrel{}{=}54\\ \frac{1}{4}{\left(6^2\right)}^{\frac{3}{2}}\stackrel{}{=}54\\ \frac{1}{4}{\left(6\right)}^3\stackrel{}{=}54\\ \frac{1}{4}\left(216\right)\stackrel{}{=}54\\ 54=54\end{array}$