Answer :
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].
### 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].
