Sure, let's evaluate the expression [tex]\((4x - 16)^{\frac{1}{2}} - 36\)[/tex] for each of the given values [tex]\(x = 5\)[/tex], [tex]\(x = 13\)[/tex], [tex]\(x = 20\)[/tex], and [tex]\(x = 328\)[/tex].
### For [tex]\( x = 5 \)[/tex]:
1. Substitute [tex]\( x = 5 \)[/tex] into the expression:
[tex]\[
(4 \cdot 5 - 16)^{\frac{1}{2}} - 36
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
(20 - 16)^{\frac{1}{2}} - 36
\][/tex]
3. Continue simplifying:
[tex]\[
4^{\frac{1}{2}} - 36
\][/tex]
4. Evaluate the square root:
[tex]\[
2 - 36
\][/tex]
5. Subtract 36 from 2:
[tex]\[
-34
\][/tex]
So, the result for [tex]\( x = 5 \)[/tex] is [tex]\( -34 \)[/tex].
### For [tex]\( x = 13 \)[/tex]:
1. Substitute [tex]\( x = 13 \)[/tex] into the expression:
[tex]\[
(4 \cdot 13 - 16)^{\frac{1}{2}} - 36
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
(52 - 16)^{\frac{1}{2}} - 36
\][/tex]
3. Continue simplifying:
[tex]\[
36^{\frac{1}{2}} - 36
\][/tex]
4. Evaluate the square root:
[tex]\[
6 - 36
\][/tex]
5. Subtract 36 from 6:
[tex]\[
-30
\][/tex]
So, the result for [tex]\( x = 13 \)[/tex] is [tex]\( -30 \)[/tex].
### For [tex]\( x = 20 \)[/tex]:
1. Substitute [tex]\( x = 20 \)[/tex] into the expression:
[tex]\[
(4 \cdot 20 - 16)^{\frac{1}{2}} - 36
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
(80 - 16)^{\frac{1}{2}} - 36
\][/tex]
3. Continue simplifying:
[tex]\[
64^{\frac{1}{2}} - 36
\][/tex]
4. Evaluate the square root:
[tex]\[
8 - 36
\][/tex]
5. Subtract 36 from 8:
[tex]\[
-28
\][/tex]
So, the result for [tex]\( x = 20 \)[/tex] is [tex]\( -28 \)[/tex].
### For [tex]\( x = 328 \)[/tex]:
1. Substitute [tex]\( x = 328 \)[/tex] into the expression:
[tex]\[
(4 \cdot 328 - 16)^{\frac{1}{2}} - 36
\][/tex]
2. Simplify inside the parentheses:
[tex]\[
(1312 - 16)^{\frac{1}{2}} - 36
\][/tex]
3. Continue simplifying:
[tex]\[
1296^{\frac{1}{2}} - 36
\][/tex]
4. Evaluate the square root (since [tex]\( 1296 \)[/tex] is a perfect square; [tex]\( 1296 = 36^2 \)[/tex]):
[tex]\[
36 - 36
\][/tex]
5. Subtract 36 from 36:
[tex]\[
0
\][/tex]
So, the result for [tex]\( x = 328 \)[/tex] is [tex]\( 0 \)[/tex].
In summary, the results for the given values of [tex]\( x \)[/tex] are:
- For [tex]\( x = 5 \)[/tex], the result is [tex]\( -34 \)[/tex].
- For [tex]\( x = 13 \)[/tex], the result is [tex]\( -30 \)[/tex].
- For [tex]\( x = 20 \)[/tex], the result is [tex]\( -28 \)[/tex].
- For [tex]\( x = 328 \)[/tex], the result is [tex]\( 0 \)[/tex].
