Sure! Let's go through the steps to simplify the expression [tex]\(\left(y^{\frac{4}{3}} \cdot y^{\frac{2}{3}}\right)^{-\frac{1}{2}}\)[/tex] step-by-step.
1. Apply the Product of Powers Property: The product of powers property states that [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. So, we first combine the exponents inside the parentheses: [tex]\[
y^{\frac{4}{3}} \cdot y^{\frac{2}{3}} = y^{\left(\frac{4}{3} + \frac{2}{3}\right)}
\][/tex]
2. Sum the exponents: Now, add [tex]\(\frac{4}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]: [tex]\[
\frac{4}{3} + \frac{2}{3} = \frac{6}{3} = 2
\][/tex] So, the expression inside the parentheses simplifies to: [tex]\[
y^2
\][/tex]
3. Apply the Power of a Power Property: Now we need to apply the exponent [tex]\(-\frac{1}{2}\)[/tex] to the simplified expression [tex]\(y^2\)[/tex]: [tex]\[
\left(y^2\right)^{-\frac{1}{2}}
\][/tex]
4. Simplify the Exponent: The power of a power property states [tex]\((a^m)^n = a^{mn}\)[/tex], so we multiply the exponents: [tex]\[
y^{2 \cdot \left(-\frac{1}{2}\right)} = y^{-1}
\][/tex]
Therefore, the expression [tex]\(\left(y^{\frac{4}{3}} \cdot y^{\frac{2}{3}}\right)^{-\frac{1}{2}}\)[/tex] simplifies to [tex]\( y^{-1} \)[/tex].
