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]
