Questions: Solve the equation. Express your answer in exact simplest form. sqrt[4]h+4 = sqrt[4]3 h-6 The solution set is □.

Solve the equation. Express your answer in exact simplest form. sqrt[4]h+4 = sqrt[4]3 h-6 The solution set is □.
Transcript text: Solve the equation. Express your answer in exact simplest form. \[ \sqrt[4]{h+4}=\sqrt[4]{3 h-6} \] The solution set is $\square$.
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Solution

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Solution Steps

To solve the equation \(\sqrt[4]{h+4} = \sqrt[4]{3h-6}\), we can start by raising both sides to the power of 4 to eliminate the fourth roots. This will give us a polynomial equation that we can solve for \(h\).

Solution Approach
  1. Raise both sides of the equation to the power of 4.
  2. Simplify the resulting polynomial equation.
  3. Solve the polynomial equation for \(h\).
  4. Verify the solutions to ensure they satisfy the original equation.
Step 1: Raise Both Sides to the Power of 4

Starting with the equation \[ \sqrt[4]{h+4} = \sqrt[4]{3h-6}, \] we raise both sides to the power of 4: \[ h + 4 = 3h - 6. \]

Step 2: Simplify the Equation

Rearranging the equation gives: \[ h + 4 + 6 = 3h, \] which simplifies to: \[ 10 = 3h - h. \] This further simplifies to: \[ 10 = 2h. \]

Step 3: Solve for \(h\)

Dividing both sides by 2, we find: \[ h = 5. \]

Step 4: Verify the Solution

To ensure that \(h = 5\) satisfies the original equation, we substitute \(h\) back into the equation: \[ \sqrt[4]{5 + 4} = \sqrt[4]{3(5) - 6}. \] Calculating both sides: \[ \sqrt[4]{9} = \sqrt[4]{9}, \] which confirms that both sides are equal.

Final Answer

The solution set is \(\boxed{h = 5}\).

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