x = –6
Solution:
Given expression is [tex]\sqrt[3]{x-2}+5=3[/tex].
Step 1: Isolate the radical by subtracting 5 from both sides of the equation.
[tex]\Rightarrow\sqrt[3]{x-2}+5-5=3-5[/tex]
[tex]\Rightarrow\sqrt[3]{x-2}=-2[/tex]
Step 2: Cube both sides of the equation to remove the cube root.
[tex]\Rightarrow(\sqrt[3]{x-2})^3=(-2)^3[/tex]
Cube and cube root get canceled in left side of the equation.
[tex]\Rightarrow\ x-2=-8[/tex]
Step 3: To solve for x.
Add 2 on both sides of the equation.
[tex]\Rightarrow\ x-2+2=-8+2[/tex]
[tex]\Rightarrow\ x=-6[/tex]
Hence the solution is x = –6.
