Question
Analyze the work below. Circle the error and then correctly solve the equation. 9. \( \sqrt[4]{6 x}+4=5 \sqrt[4]{6 x}+12 \) Student Work: \( \sqrt[4]{6 x}+4=5 \sqrt[4]{6 x}+12 \) \( 6 \sqrt[4]{6 x}+4=12 \) \( 6 \sqrt[4]{6 x}=8 \) \( \sqrt[4]{6 x}=\frac{4}{3} \) \( 6 x=\frac{256}{81} \) \( x=\frac{128}{243} \)
Ask by Colon Olson. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Error in Step 2:** The student incorrectly transformed the equation by adding \(6\sqrt[4]{6x}\) instead of isolating the \(\sqrt[4]{6x}\) terms properly.
**Correct Solution:**
1. Start with the original equation:
\[
\sqrt[4]{6x} + 4 = 5\sqrt[4]{6x} + 12
\]
2. Subtract \(\sqrt[4]{6x}\) from both sides:
\[
4 = 4\sqrt[4]{6x} + 12
\]
3. Subtract 12 from both sides:
\[
-8 = 4\sqrt[4]{6x}
\]
4. Divide both sides by 4:
\[
\sqrt[4]{6x} = -2
\]
5. **Conclusion:** Since the fourth root of a real number cannot be negative, there is **no real solution** to the equation.
**Final Answer:** There is no real solution to the equation.
Solution
**Error Identification:**
The error occurs in **Step 2** of the student's work. The student incorrectly transforms the original equation:
\[
\sqrt[4]{6x} + 4 = 5\sqrt[4]{6x} + 12
\]
into:
\[
6\sqrt[4]{6x} + 4 = 12
\]
This transformation is incorrect because it doesn't properly isolate the \(\sqrt[4]{6x}\) terms.
**Correct Solution:**
Let's solve the equation step-by-step correctly.
1. **Original Equation:**
\[
\sqrt[4]{6x} + 4 = 5\sqrt[4]{6x} + 12
\]
2. **Isolate the \(\sqrt[4]{6x}\) Terms:**
Subtract \(\sqrt[4]{6x}\) from both sides:
\[
4 = 4\sqrt[4]{6x} + 12
\]
3. **Move Constants to One Side:**
Subtract 12 from both sides:
\[
-8 = 4\sqrt[4]{6x}
\]
4. **Solve for \(\sqrt[4]{6x}\):**
Divide both sides by 4:
\[
\sqrt[4]{6x} = -2
\]
5. **Analyze the Solution:**
The fourth root of a real number cannot be negative. Therefore, there's **no real solution** to the equation.
**Conclusion:**
The student's error in Step 2 led to an incorrect solution. The correct approach shows that the equation has **no real solution**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
In the student work, the error lies in the step where they combined terms incorrectly. The correct manipulation should be: First, subtract \(\sqrt[4]{6 x}\) from both sides: \[ 4 = 4 \sqrt[4]{6 x} + 12 \] Subtract \(12\) from both sides: \[ 4 - 12 = 4 \sqrt[4]{6 x} \] This leads to: \[ -8 = 4 \sqrt[4]{6 x} \] Now dividing both sides by \(4\): \[ -2 = \sqrt[4]{6 x} \] Since the fourth root cannot be negative, this indicates there is no solution to the original equation. x has no valid value because a fourth root can never yield a negative result!
