Chapter 6.7, Problem 49PPS

To determine

To Solve:

  ${\left(2x-1\right)}^{\frac{1}{4}}-2=1$

Answer to Problem 49PPS

F is the correct option.

Explanation of Solution

Given Information:

Given equation:

  ${\left(2x-1\right)}^{\frac{1}{4}}-2=1$

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 ${\left(2x-1\right)}^{\frac{1}{4}}-2=1$

$\begin{array}{l}{\left(2x-1\right)}^{\frac{1}{4}}-2=1\\ \Rightarrow {\left(2x-1\right)}^{\frac{1}{4}}=1+\mathrm{2............}[Add\text{ 2 each side]}\\ \Rightarrow {\left(2x-1\right)}^{\frac{1}{4}}=3\\ \Rightarrow {\left({\left(2x-1\right)}^{\frac{1}{4}}\right)}^4=3^4\mathrm{.............}[\text{Raise each side to the power 4=index]}\\ \Rightarrow 2x-1=81\\ \Rightarrow 2x=81+\mathrm{1.................}[Add\text{ 1 each side]}\\ \Rightarrow 2x=82\\ \Rightarrow \frac{2x}{2}=\frac{82}{2}\mathrm{.....................}[\text{Divide each side by 2]}\\ \Rightarrow x=41\end{array}$

So, correct option is F.