To solve the expression [tex]\(\sqrt[5]{\log _2\left(\frac{3 x^3}{2}\right)\left(\frac{1}{x}\right)^3}\)[/tex], let's go through it step-by-step and simplify it.
1. Rewrite the expression:
[tex]\[
\sqrt[5]{\log_2 \left( \frac{3x^3}{2} \right) \left( \frac{1}{x} \right)^3}
\][/tex]
The exponent [tex]\( \left( \frac{1}{x} \right)^3 \)[/tex] simplifies to [tex]\(\frac{1}{x^3}\)[/tex].
2. Combine the terms inside the logarithm:
[tex]\[
\log_2 \left( \frac{3x^3}{2} \right) \cdot \frac{1}{x^3}
\][/tex]
3. Distribute the fraction:
[tex]\[
\left( \log_2 \left( \frac{3x^3}{2} \right) \right) \cdot \frac{1}{x^3}
\][/tex]
4. Combine into a single logarithm divided by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{\log_2 \left( \frac{3x^3}{2} \right)}{x^3}
\][/tex]
5. Now take the fifth root of the entire fraction:
[tex]\[
\left( \frac{\log_2 \left( \frac{3x^3}{2} \right)}{x^3} \right)^{\frac{1}{5}}
\][/tex]
6. Rewrite using exponents and roots:
[tex]\[
\left( \frac{\log_2 \left( \frac{3x^3}{2} \right)}{x^3} \right)^{1/5}
\][/tex]
7. Simplification with properties of logarithms and exponents:
Notice that the expression under the logarithm [tex]\(\frac{3x^3}{2}\)[/tex] can be simplified into parts, but the given result should be treated as follows:
[tex]\[
(\log(3*x^3/2)/x^3)^{0.2}/\log(2)^{0.2}
\][/tex]
Here, [tex]\(0.2\)[/tex] is the fractional exponent representing the [tex]\(1/5\)[/tex] power.
So, the final simplified answer is:
[tex]\[
\left( \frac{\log(3*x^3/2)}{x^3} \right)^{0.2} / (\log(2))^{0.2}
\][/tex]
This answer explains the step-by-step simplification for the expression [tex]\(\sqrt[5]{\log _2\left(\frac{3 x^3}{2}\right)\left(\frac{1}{x}\right)^3}\)[/tex].
