To simplify the expression [tex]\(\sqrt[3]{\frac{x^3 (y^4)^3}{6^3}}\)[/tex], follow these steps:
Step 1: Simplify the expression inside the cube root.
[tex]\[
(y^4)^3 \text{ can be written as } y^{4 \cdot 3} = y^{12}
\][/tex]
So, the expression inside the cube root becomes:
[tex]\[
\frac{x^3 y^{12}}{6^3}
\][/tex]
Step 2: We now need to find the cube root of the entire expression:
[tex]\[
\sqrt[3]{\frac{x^3 y^{12}}{6^3}}
\][/tex]
Step 3: Recall the property of cube roots where [tex]\(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\)[/tex]. Applying this property:
[tex]\[
\sqrt[3]{\frac{x^3 y^{12}}{6^3}} = \frac{\sqrt[3]{x^3 y^{12}}}{\sqrt[3]{6^3}}
\][/tex]
Step 4: Simplify the cube root of the numerator and the denominator separately. Starting with the numerator:
[tex]\[
\sqrt[3]{x^3 y^{12}}
\][/tex]
Using properties of exponents and cube roots, we know [tex]\(\sqrt[3]{a^3} = a\)[/tex]. Therefore:
[tex]\[
\sqrt[3]{x^3 y^{12}} = x y^{12/3} = x y^4
\][/tex]
Step 5: Simplify the cube root of the denominator:
[tex]\[
\sqrt[3]{6^3} = 6
\][/tex]
Step 6: Combine these results:
[tex]\[
\frac{xy^4}{6}
\][/tex]
Putting it all together, the simplified expression is:
[tex]\[
\frac{(xy^4)}{6}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[3]{\frac{x^3 (y^4)^3}{6^3}}\)[/tex] is:
[tex]\[
\frac{(x^3 y^{12})^{1/3}}{6}
\][/tex]
or equivalently:
[tex]\[
\frac{(x y^4)}{6}
\][/tex]
