What is the solution of [tex][tex]$(4x - 16)^{\frac{1}{2}} = 36$[/tex][/tex]?

A. [tex][tex]$x = 5$[/tex][/tex]
B. [tex][tex]$x = 13$[/tex][/tex]
C. [tex][tex]$x = 20$[/tex][/tex]
D. [tex][tex]$x = 328$[/tex][/tex]



Answer :

To solve the equation [tex]\((4x - 16)^{\frac{1}{2}} = 36\)[/tex], we need to go through the following steps:

1. Square both sides of the equation to eliminate the square root:

[tex]\[ \left( (4x - 16)^{\frac{1}{2}} \right)^2 = 36^2 \][/tex]

This simplifies to:

[tex]\[ 4x - 16 = 1296 \][/tex]

2. Isolate the term involving [tex]\(x\)[/tex]:

To do this, add 16 to both sides of the equation:

[tex]\[ 4x - 16 + 16 = 1296 + 16 \][/tex]

Which simplifies to:

[tex]\[ 4x = 1312 \][/tex]

3. Solve for [tex]\(x\)[/tex]:

Divide both sides by 4:

[tex]\[ x = \frac{1312}{4} \][/tex]

Simplifying the division:

[tex]\[ x = 328 \][/tex]

4. Verify the solution:

Substitute [tex]\(x = 328\)[/tex] back into the original equation to ensure both sides are equal:

[tex]\[ (4 \cdot 328 - 16)^{\frac{1}{2}} = 36 \][/tex]

Simplifying inside the parentheses:

[tex]\[ (1312 - 16)^{\frac{1}{2}} = 36 \][/tex]

Which becomes:

[tex]\[ (1296)^{\frac{1}{2}} = 36 \][/tex]

Since [tex]\(1296\)[/tex] is the square of [tex]\(36\)[/tex] ([tex]\(36^2 = 1296\)[/tex]), taking the square root of [tex]\(1296\)[/tex] indeed gives [tex]\(36\)[/tex]:

[tex]\[ 36 = 36 \][/tex]

Thus, the solution to the equation [tex]\((4x - 16)^{\frac{1}{2}} = 36\)[/tex] is [tex]\(x = 328\)[/tex].

Therefore, the correct answer is:
[tex]\[ x = 328 \][/tex]