To rewrite the expression [tex]\(\left(4 \sqrt[4]{y^5}\right)^{\frac{1}{2}}\)[/tex] in the form [tex]\(k \cdot y^n\)[/tex], follow these steps:
1. Rewrite the fourth root of [tex]\(y^5\)[/tex] as an exponent:
[tex]\[
\sqrt[4]{y^5} = y^{\frac{5}{4}}
\][/tex]
2. Substitute this back into the original expression:
[tex]\[
\left(4 \cdot y^{\frac{5}{4}}\right)^{\frac{1}{2}}
\][/tex]
3. Apply the power of a product rule:
[tex]\[
(ab)^c = a^c \cdot b^c
\][/tex]
This means:
[tex]\[
\left(4 \cdot y^{\frac{5}{4}}\right)^{\frac{1}{2}} = 4^{\frac{1}{2}} \cdot \left(y^{\frac{5}{4}}\right)^{\frac{1}{2}}
\][/tex]
4. Simplify each part separately:
- For [tex]\(4^{\frac{1}{2}}\)[/tex]:
[tex]\[
4^{\frac{1}{2}} = \sqrt{4} = 2
\][/tex]
- For [tex]\(\left(y^{\frac{5}{4}}\right)^{\frac{1}{2}}\)[/tex]:
[tex]\[
\left(y^{\frac{5}{4}}\right)^{\frac{1}{2}} = y^{\left(\frac{5}{4} \cdot \frac{1}{2}\right)} = y^{\frac{5}{8}}
\][/tex]
5. Combine the results:
[tex]\[
2 \cdot y^{\frac{5}{8}}
\][/tex]
Thus, the expression [tex]\(\left(4 \sqrt[4]{y^5}\right)^{\frac{1}{2}}\)[/tex] can be rewritten in the form [tex]\(k \cdot y^n\)[/tex] as:
[tex]\[
2 \cdot y^{\frac{5}{8}}
\][/tex]
