Chapter 6.7, Problem 46PPS

To determine

To Solve:

  $\frac{1}{7}{\left(14a\right)}^{\frac{1}{3}}=1$

Answer to Problem 46PPS

  $a=24.5$

Explanation of Solution

Given Information:

Given equation:

  $\frac{1}{7}{\left(14a\right)}^{\frac{1}{3}}=1$

Concept Used :

Solve Radical equations:

• Isolate the radical on one side of the equation.
• Raise each side of the equation to a power equal to the index of the radical to eliminate the radical .
• Solve the resulting polynomial equation.Check your results.
• When solving radical equations, the result may be a number that does not satisfy the original equation . Such a number is called an extraneous solution.

Calculation:

Given $\frac{1}{7}{\left(14a\right)}^{\frac{1}{3}}=1$

$\begin{array}{l}\frac{1}{7}{\left(14a\right)}^{\frac{1}{3}}=1\\ \Rightarrow {\left(14a\right)}^{\frac{1}{3}}=1(7)\mathrm{.....................}[\text{Multiply each side by 7]}\\ \Rightarrow {\left(14a\right)}^{\frac{1}{3}}=7\\ $ \Rightarrow {\left({\left(14a\right)}^{\frac{1}{3}}\right)}^3={(7)}^3\mathrm{...............}[\text{Raise each side to the power 3=index]}$\Rightarrow 14a=343\\ \Rightarrow \frac{14a}{14}=\frac{343}{14}\mathrm{..............................}[\text{Divide each side by 14]}\\ \Rightarrow a=24.5\end{array}$

So, the solution is $a=24.5$