To simplify the expression [tex]\(\frac{\left(2 \pi y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{2}{3}}}\)[/tex], we will follow these steps:
1. Simplify the Numerator:
Consider the term [tex]\(\left(2 \pi y^{-2}\right)^{\frac{1}{3}}\)[/tex]:
[tex]\[
\left(2 \pi y^{-2}\right)^{\frac{1}{3}}
\][/tex]
We can distribute the exponent [tex]\(\frac{1}{3}\)[/tex] to each part inside the parentheses:
[tex]\[
= \left(2 \pi\right)^{\frac{1}{3}} \cdot \left(y^{-2}\right)^{\frac{1}{3}}
\][/tex]
Recall the exponent rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
= \left(2 \pi\right)^{\frac{1}{3}} \cdot y^{- \frac{2}{3}}
\][/tex]
Therefore, the numerator simplifies to:
[tex]\[
\left(2 \pi\right)^{\frac{1}{3}} y^{- \frac{2}{3}}
\][/tex]
2. Simplify the Denominator:
The denominator is already simplified:
[tex]\[
y^{- \frac{2}{3}}
\][/tex]
3. Combine the Simplified Numerator and Denominator:
Now, we substitute the simplified forms of the numerator and the denominator back into the fraction:
[tex]\[
\frac{\left(2 \pi\right)^{\frac{1}{3}} y^{- \frac{2}{3}}}{y^{- \frac{2}{3}}}
\][/tex]
4. Cancel Common Terms:
Notice that [tex]\(y^{- \frac{2}{3}}\)[/tex] in the numerator and the denominator can cancel each other out:
[tex]\[
= \left(2 \pi\right)^{\frac{1}{3}}
\][/tex]
The simplified expression is:
[tex]\[
\boxed{\left(2 \pi\right)^{\frac{1}{3}}}
\][/tex]
