Questions: Find the real solutions of the following equation. (4 x-4)^(1/3)=-4

Find the real solutions of the following equation. (4 x-4)^(1/3)=-4
Transcript text: Find the real solutions of the following equation. \[ (4 x-4)^{\frac{1}{3}}=-4 \]
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Solution

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Solution Steps

To find the real solutions of the given equation \((4x - 4)^{\frac{1}{3}} = -4\), we need to isolate \(x\). We can start by cubing both sides of the equation to eliminate the cube root, and then solve for \(x\).

Step 1: Cube Both Sides of the Equation

To eliminate the cube root, we cube both sides of the equation: \[ \left( (4x - 4)^{\frac{1}{3}} \right)^3 = (-4)^3 \] This simplifies to: \[ 4x - 4 = -64 \]

Step 2: Solve for \(x\)

Next, we solve for \(x\) by isolating it on one side of the equation: \[ 4x - 4 = -64 \] Add 4 to both sides: \[ 4x = -60 \] Divide both sides by 4: \[ x = -15 \]

Final Answer

\(\boxed{x = -15}\)

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