To solve the equation [tex]\(\sqrt[3]{x + 8} = -4\)[/tex], we need to follow a series of mathematical steps in order to isolate and solve for [tex]\(x\)[/tex]. Let's go through these steps systematically:
1. Understand the equation: The given equation is [tex]\(\sqrt[3]{x + 8} = -4\)[/tex].
2. Remove the cube root: To eliminate the cube root, cube both sides of the equation.
[tex]\[
\left( \sqrt[3]{x + 8} \right)^3 = (-4)^3
\][/tex]
This simplifies to:
[tex]\[
x + 8 = (-4)^3
\][/tex]
3. Calculate the power on the right side: Evaluate [tex]\((-4)^3\)[/tex]:
[tex]\[
(-4)^3 = -64
\][/tex]
4. Set the equation in a solvable form: Substitute [tex]\(-64\)[/tex] for [tex]\((-4)^3\)[/tex] in the equation:
[tex]\[
x + 8 = -64
\][/tex]
5. Isolate [tex]\(x\)[/tex]: Subtract 8 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -64 - 8
\][/tex]
[tex]\[
x = -72
\][/tex]
Therefore, the correct solution is [tex]\(x = -72\)[/tex].
To verify, substitute [tex]\(x = -72\)[/tex] back into the original equation:
[tex]\[
\sqrt[3]{-72 + 8} = \sqrt[3]{-64} = -4
\][/tex]
The left side equals the right side, confirming our solution is correct.
Thus, the solution to [tex]\(\sqrt[3]{x + 8} = -4\)[/tex] is:
[tex]\[
x = -72
\][/tex]
