To solve the expression [tex]\(\left(\sqrt[3]{x^2} \cdot \sqrt[6]{x^4}\right)^{-3}\)[/tex], we'll break it down step by step:
1. Simplify the expressions inside the parentheses:
First, we'll address [tex]\(\sqrt[3]{x^2}\)[/tex] and [tex]\(\sqrt[6]{x^4}\)[/tex] using their equivalent fractional exponents.
[tex]\[
\sqrt[3]{x^2} = x^{\frac{2}{3}}
\][/tex]
[tex]\[
\sqrt[6]{x^4} = x^{\frac{4}{6}} = x^{\frac{2}{3}}
\][/tex]
Therefore, the expression inside the parentheses becomes:
[tex]\[
\left(x^{\frac{2}{3}} \cdot x^{\frac{2}{3}}\right)
\][/tex]
2. Combine the exponents:
When multiplying expressions with the same base, you add the exponents:
[tex]\[
x^{\frac{2}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{2}{3} + \frac{2}{3}} = x^{\frac{4}{3}}
\][/tex]
So the expression now is:
[tex]\[
\left(x^{\frac{4}{3}}\right)^{-3}
\][/tex]
3. Distribute the outer exponent:
When raising a power to another power, you multiply the exponents:
[tex]\[
\left(x^{\frac{4}{3}}\right)^{-3} = x^{\left(\frac{4}{3}\right) \cdot (-3)}
\][/tex]
Multiplying the exponents:
[tex]\[
x^{\frac{4}{3} \cdot -3} = x^{-\frac{12}{3}} = x^{-4}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\left(\sqrt[3]{x^2} \cdot \sqrt[6]{x^4}\right)^{-3}\)[/tex] is:
[tex]\[
x^{-4}
\][/tex]
This is the final answer.
