To simplify the expression [tex]\(\frac{\left(27 y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{1}{3}}}\)[/tex], follow these steps:
1. Let's start with the numerator [tex]\((27 y^{-2})^{1/3}\)[/tex].
- [tex]\(27\)[/tex] can be expressed as [tex]\(3^3\)[/tex], thus:
[tex]\[
(27 y^{-2})^{1/3} = \left( 3^3 y^{-2} \right)^{1/3}
\][/tex]
- Using the power rule [tex]\((a^m \cdot b^n)^{k} = a^{mk} \cdot b^{nk}\)[/tex], we get:
[tex]\[
(3^3 y^{-2})^{1/3} = 3^{3 \cdot 1/3} \cdot y^{-2 \cdot 1/3} = 3 \cdot y^{-2/3}
\][/tex]
2. Now, let's look at the denominator [tex]\(y^{-1/3}\)[/tex].
3. Combining the simplified numerator and denominator, we have:
[tex]\[
\frac{3 y^{-2/3}}{y^{-1/3}}
\][/tex]
4. To simplify this division of exponents, recall the property of exponents [tex]\(a^{m}/a^{n} = a^{m-n}\)[/tex]:
- Thus, we can write:
[tex]\[
3 \cdot y^{(-2/3) - (-1/3)} = 3 \cdot y^{(-2/3 + 1/3)} = 3 \cdot y^{-1/3}
\][/tex]
So, the expression [tex]\(\frac{\left(27 y^{-2}\right)^{\frac{1}{3}}}{y^{-\frac{1}{3}}}\)[/tex] simplifies to:
[tex]\[
3y^{-1/3}
\][/tex]
To express the final answer in the required form without negative exponents:
[tex]\[
3 \cdot y^{1/3}
\][/tex]
The simplified expression is [tex]\( 3 y^{1/3} \)[/tex].
