Alright, let's solve each of these radical equations step-by-step to find the correct matches.
### Step-by-Step Solution:
1. Equation: [tex]\( \sqrt{(x-1)^3} = 8 \)[/tex]
- To isolate [tex]\( x \)[/tex]:
[tex]\[
(x-1)^3 = 8^2 \\
(x-1)^3 = 64 \\
x - 1 = 64^{1/3} \\
x - 1 = 4 \\
x = 5
\][/tex]
- Therefore, [tex]\( x = 5 \)[/tex] for [tex]\( \sqrt{(x-1)^3} = 8 \)[/tex].
2. Equation: [tex]\( \sqrt[4]{(x-3)^5} = 32 \)[/tex]
- To isolate [tex]\( x \)[/tex]:
[tex]\[
(x-3)^5 = 32^4 \\
(x-3)^5 = 1048576 \\
x - 3 = 1048576^{1/5} \\
x - 3 \approx 16.34 \\
x \approx 19.34
\][/tex]
- However, there is no matching solution for [tex]\( x = 19.34 \)[/tex] given in the set.
3. Equation: [tex]\( \sqrt{(x-4)^3} = 125 \)[/tex]
- To isolate [tex]\( x \)[/tex]:
[tex]\[
(x-4)^3 = 125^2 \\
(x-4)^3 = 15625 \\
x - 4 = 15625^{1/3} \\
x - 4 \approx 25 \\
x \approx 29
\][/tex]
- Therefore, [tex]\( x = 29 \)[/tex] for [tex]\( \sqrt{(x-4)^3} = 125 \)[/tex].
4. Equation: [tex]\( \sqrt[3]{(x+2)^4} = 16 \)[/tex]
- To isolate [tex]\( x \)[/tex]:
[tex]\[
(x+2)^4 = 16^3 \\
(x+2)^4 = 4096 \\
x + 2 = 4096^{1/4} \\
x + 2 = 8 \\
x = 6
\][/tex]
- Therefore, [tex]\( x = 6 \)[/tex] for [tex]\( \sqrt[3]{(x+2)^4} = 16 \)[/tex].
### Matching Equations to Solutions:
- [tex]\( \sqrt{(x-1)^3} = 8 \)[/tex] corresponds to [tex]\( x = 5 \)[/tex].
- [tex]\( \sqrt{(x-4)^3} = 125 \)[/tex] corresponds to [tex]\( x = 29 \)[/tex].
- The remaining equation ([tex]\( \sqrt[3]{(x+2)^4} = 16 \)[/tex]) corresponds to [tex]\( x = 6 \)[/tex].
Therefore, the matches are:
- [tex]\( x = 5 \)[/tex] [tex]\(\longrightarrow\)[/tex] [tex]\( \sqrt{(x-1)^3} = 8 \)[/tex]
- [tex]\( x = 29 \)[/tex] [tex]\(\longrightarrow\)[/tex] [tex]\( \sqrt{(x-4)^3} = 125 \)[/tex]
