Question
Find the positive solution of the equation. \[ 3 x^{\frac{4}{3}}-4=239 \]
Ask by Phillips Reeves. in the United States
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The positive solution is 27.
Solution
To solve the equation:
\[
3 x^{\frac{4}{3}} - 4 = 239
\]
follow these steps:
1. **Isolate the term with \( x \):**
\[
3 x^{\frac{4}{3}} = 239 + 4
\]
\[
3 x^{\frac{4}{3}} = 243
\]
2. **Divide both sides by 3:**
\[
x^{\frac{4}{3}} = \frac{243}{3}
\]
\[
x^{\frac{4}{3}} = 81
\]
3. **Solve for \( x \) by raising both sides to the reciprocal exponent \( \frac{3}{4} \):**
\[
x = 81^{\frac{3}{4}}
\]
4. **Simplify \( 81^{\frac{3}{4}} \):**
- Recognize that \( 81 = 3^4 \).
- Substitute and simplify:
\[
81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^{4 \times \frac{3}{4}} = 3^3 = 27
\]
**Therefore, the positive solution is \( \boxed{27} \).**
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Extra Insights
To find the positive solution of the equation \[ 3 x^{\frac{4}{3}} - 4 = 239, \] first, we can isolate the term involving \( x \): \[ 3 x^{\frac{4}{3}} = 239 + 4, \] which simplifies to \[ 3 x^{\frac{4}{3}} = 243. \] Next, divide both sides by 3: \[ x^{\frac{4}{3}} = \frac{243}{3} = 81. \] Now, to solve for \( x \), we can raise both sides to the power of \( \frac{3}{4} \): \[ x = 81^{\frac{3}{4}}. \] Calculating \( 81^{\frac{3}{4}} \), we first find that \( 81 = 3^4 \), so: \[ 81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27. \] Thus, the positive solution to the equation is \[ \boxed{27}. \]
