Sure, let's solve each radical equation step-by-step.
### Equation 1: [tex]\(\sqrt{(x-1)^3} = 8\)[/tex]
1. To eliminate the square root, square both sides of the equation:
[tex]\[
(\sqrt{(x-1)^3})^2 = 8^2
\][/tex]
2. This simplifies to:
[tex]\[
(x-1)^3 = 64
\][/tex]
3. To solve for [tex]\(x\)[/tex], take the cube root of both sides:
[tex]\[
x - 1 = \sqrt[3]{64}
\][/tex]
4. Simplify the cube root:
[tex]\[
x - 1 = 4
\][/tex]
5. Finally, solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4 + 1
\][/tex]
[tex]\[
x = 5.0
\][/tex]
### Equation 2: [tex]\(\sqrt[4]{(x-3)^5} = 32\)[/tex]
1. To eliminate the fourth root, raise both sides of the equation to the power of 4:
[tex]\[
(\sqrt[4]{(x-3)^5})^4 = 32^4
\][/tex]
2. This simplifies to:
[tex]\[
(x-3)^5 = 1048576
\][/tex]
3. To solve for [tex]\(x\)[/tex], take the fifth root of both sides:
[tex]\[
x - 3 = \sqrt[5]{1048576}
\][/tex]
4. Simplify the fifth root:
[tex]\[
x - 3 = 16
\][/tex]
5. Finally, solve for [tex]\(x\)[/tex]:
[tex]\[
x = 16 + 3
\][/tex]
[tex]\[
x = 19.000000000000004
\][/tex]
### Equation 3: [tex]\(\sqrt[3]{(x+2)^4} = 16\)[/tex]
1. To eliminate the cube root, raise both sides of the equation to the power of 3:
[tex]\[
(\sqrt[3]{(x+2)^4})^3 = 16^3
\][/tex]
2. This simplifies to:
[tex]\[
(x+2)^4 = 4096
\][/tex]
3. To solve for [tex]\(x\)[/tex], take the fourth root of both sides:
[tex]\[
x + 2 = \sqrt[4]{4096}
\][/tex]
4. Simplify the fourth root:
[tex]\[
x + 2 = 6
\][/tex]
5. Finally, solve for [tex]\(x\)[/tex]:
[tex]\[
x = 6 - 2
\][/tex]
[tex]\[
x = 4
\][/tex]
### Preset value:
- Given:
[tex]\[
x = 19
\][/tex]
### Summary of solutions:
Therefore, the solutions to the given equations are:
[tex]\[
x_1 = 5.0, \quad x_2 = 19.000000000000004, \quad x_3 = 4, \quad \text{and} \quad x = 19
\][/tex]
