Question
Solve for x, frac{4 x^{5}-2}{2}-frac{8+3 x^{5}}{3}=frac{6 x^{5}-1}{5}
Ask by Ellis Long.
Mar 06,2025 18:06
UpStudy AI Solution
Tutor-Verified Answer
Answer
\( x = -\sqrt[5]{\frac{52}{3}} \)
Solution
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Bonus Knowledge
To solve for \( x^{\prime \prime} \) in the given equation, let's start by simplifying the left-hand side and combining the fractions:
1. Start with the equation:
\[
\frac{4 x^{5}-2}{2}-\frac{8+3 x^{5}}{3}=\frac{6 x^{5}-1}{5}
\]
2. Simplify:
\[
\frac{4 x^{5}}{2} - \frac{2}{2} - \frac{8}{3} - \frac{3 x^{5}}{3} = \frac{6 x^{5}}{5} - \frac{1}{5}
\]
\[
2 x^{5} - 1 - \frac{8 + 3 x^{5}}{3} = \frac{6 x^{5} - 1}{5}
\]
3. Now find a common denominator for the left-hand side, which is 3:
\[
\frac{6 x^{5} - 3}{3} - \frac{8 + 3 x^{5}}{3} = \frac{6 x^{5} - 1}{5}
\]
\[
\frac{6 x^{5} - 3 - 8 - 3 x^{5}}{3} = \frac{6 x^{5} - 1}{5}
\]
\[
\frac{3 x^{5} - 11}{3} = \frac{6 x^{5} - 1}{5}
\]
4. Now cross-multiply to eliminate the fractions:
\[
5(3 x^{5} - 11) = 3(6 x^{5} - 1)
\]
\[
15 x^{5} - 55 = 18 x^{5} - 3
\]
5. Rearranging gives:
\[
-55 + 3 = 18 x^{5} - 15 x^{5}
\]
\[
-52 = 3 x^{5}
\]
\[
x^{5} = -\frac{52}{3}
\]
6. Finally, take the fifth root to find \( x \):
\[
x = \left(-\frac{52}{3}\right)^{\frac{1}{5}}
\]
Thus, \( x = \sqrt[5]{-\frac{52}{3}} \).
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