To solve the equation [tex]\(\sqrt[3]{-6-x} = \sqrt[3]{3x + 26}\)[/tex], we need to follow these steps:
### Step 1: Eliminate the Cube Roots
To eliminate the cube roots on both sides of the equation, we'll cube both sides. This will remove the cube roots:
[tex]\[
\left( \sqrt[3]{-6 - x} \right)^3 = \left( \sqrt[3]{3x + 26} \right)^3
\][/tex]
### Step 2: Simplify the Resulting Equation
Cubic of a cube root gives us the original value inside the root, so:
[tex]\[
-6 - x = 3x + 26
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Let's solve for [tex]\(x\)[/tex] by first isolating [tex]\(x\)[/tex]:
[tex]\[
-6 - x = 3x + 26
\][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[
-6 = 4x + 26
\][/tex]
Subtract 26 from both sides:
[tex]\[
-6 - 26 = 4x
\][/tex]
Simplify the left side:
[tex]\[
-32 = 4x
\][/tex]
Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-32}{4} = -8
\][/tex]
### Step 4: Verify the Solution
To ensure that our solution [tex]\( x = -8 \)[/tex] is correct, substitute it back into the original equation:
For the left side:
[tex]\[
\sqrt[3]{-6 - (-8)} = \sqrt[3]{-6 + 8} = \sqrt[3]{2}
\][/tex]
For the right side:
[tex]\[
\sqrt[3]{3(-8) + 26} = \sqrt[3]{-24 + 26} = \sqrt[3]{2}
\][/tex]
Since both sides are equal, the solution [tex]\( x = -8 \)[/tex] satisfies the original equation.
### Conclusion:
The solution to the equation [tex]\(\sqrt[3]{-6 - x} = \sqrt[3]{3x + 26}\)[/tex] is [tex]\( \boxed{-8} \)[/tex].
