Given the equation
[tex]\begin{gathered} 3\text{ }\sqrt[5]{(x+2)^3\text{ }}\text{ + 3 = 27} \\ \end{gathered}[/tex]Subtract 3 from both sides
[tex]\begin{gathered} 3\text{ }\sqrt[5]{(x+2)^3\text{ }}\text{ + 3-3 = 27}-3 \\ \\ 3\text{ }\sqrt[5]{(x+2)^3\text{ }}\text{ = }24 \end{gathered}[/tex]Divide both sides by 3
[tex]\begin{gathered} \text{ }\sqrt[5]{(x+2)^3\text{ }}\text{ = }\frac{24}{3} \\ \text{ }\sqrt[5]{(x+2)^3\text{ }}\text{ = 8} \\ \end{gathered}[/tex]Raise both sides to power 5 to remove the 5th root on the left hand side
[tex]\begin{gathered} \text{ (}\sqrt[5]{(x+2)^3\text{ }})^5=8^5 \\ \\ (x+2)^3=(8^{})^5 \\ (x+2)^3=\text{ 32768} \\ \end{gathered}[/tex]Take the cube root of both sides
[tex]\begin{gathered} \sqrt[3]{(x+2)^3}^{}=\sqrt[3]{32768} \\ (x+2\text{ )= 32} \\ \end{gathered}[/tex]Subtract 2 from both sides
x + 2 = 32
x = 32 - 2
x = 30
The solution to the equation is x = 30
