To rewrite the expression [tex]\(3 y^{-\frac{4}{3}} \cdot 2 \sqrt[3]{y}\)[/tex] in the form [tex]\(k \cdot y^n\)[/tex], we start by breaking down and simplifying each component of the expression.
1. Rewrite each component in terms of an exponent:
The expression [tex]\(3 y^{-\frac{4}{3}}\)[/tex] is already in exponential form.
The term [tex]\(2 \sqrt[3]{y}\)[/tex] can be rewritten using exponents as [tex]\(2 y^{\frac{1}{3}}\)[/tex].
2. Combine the coefficients:
Multiply the numerical coefficients together:
[tex]\[
3 \cdot 2 = 6
\][/tex]
3. Combine the exponents:
Add the exponents for [tex]\(y\)[/tex]:
[tex]\[
y^{-\frac{4}{3}} \cdot y^{\frac{1}{3}}
\][/tex]
Recall that when multiplying terms with the same base, the exponents are added:
[tex]\[
-\frac{4}{3} + \frac{1}{3} = -\frac{4}{3} + \frac{1}{3} = -\frac{3}{3} = -1
\][/tex]
4. Simplified expression:
Putting it all together, we get:
[tex]\[
3 y^{-\frac{4}{3}} \cdot 2 y^{\frac{1}{3}} = 6 y^{-1}
\][/tex]
In another form, [tex]\(y^{-1}\)[/tex] is the same as [tex]\(\frac{1}{y}\)[/tex].
So, the expression can be rewritten as:
[tex]\[
6 y^{-1}
\][/tex]
or equivalently,
[tex]\[
\frac{6}{y}
\][/tex]
