Questions: Solve the equation. Express your answer in exact simplest form. sqrt[3]x-2-4=1 The solution set is.

Solve the equation. Express your answer in exact simplest form. sqrt[3]x-2-4=1 The solution set is.
Transcript text: Solve the equation. Express your answer in exact simplest form. \[ \sqrt[3]{x-2}-4=1 \] The solution set is $\square$.
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Solution

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Solution Steps

To solve the equation \(\sqrt[3]{x-2} - 4 = 1\), we need to isolate the cube root term and then eliminate the cube root by cubing both sides of the equation. Finally, solve for \(x\).

Step 1: Isolate the Cube Root Term

Start with the equation: \[ \sqrt[3]{x-2} - 4 = 1 \] Add 4 to both sides to isolate the cube root term: \[ \sqrt[3]{x-2} = 5 \]

Step 2: Eliminate the Cube Root

Cube both sides of the equation to eliminate the cube root: \[ x - 2 = 5^3 \] Calculate \(5^3\): \[ x - 2 = 125 \]

Step 3: Solve for \(x\)

Add 2 to both sides to solve for \(x\): \[ x = 125 + 2 \] \[ x = 127 \]

Final Answer

The solution set is \(\boxed{127}\).

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