Let's simplify the expression [tex]\(\sqrt[3]{4xy^5} \times \sqrt[3]{2x^2y}\)[/tex].
1. Combine the Cube Roots:
Since both expressions are under cube roots, we can multiply the radicands (the expressions inside the roots) under a single cube root:
[tex]\[
\sqrt[3]{4xy^5} \times \sqrt[3]{2x^2y} = \sqrt[3]{(4xy^5) \times (2x^2y)}
\][/tex]
2. Multiply the Radicands:
Multiply the expressions inside the cube roots:
[tex]\[
\sqrt[3]{4xy^5 \cdot 2x^2y} = \sqrt[3]{(4 \cdot 2) \cdot (x \cdot x^2) \cdot (y^5 \cdot y)}
\][/tex]
Simplify:
[tex]\[
4 \cdot 2 = 8
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
[tex]\[
y^5 \cdot y = y^{5+1} = y^6
\][/tex]
Now we have:
[tex]\[
\sqrt[3]{8x^3y^6}
\][/tex]
3. Simplify Inside the Cube Root:
Notice that each part of [tex]\(8x^3y^6\)[/tex] is a perfect cube:
[tex]\[
8 = 2^3, \quad x^3 = (x)^3, \quad y^6 = (y^2)^3
\][/tex]
Therefore:
[tex]\[
8x^3y^6 = (2 \cdot x \cdot y^2)^3
\][/tex]
So now we can rewrite the expression under the cube root:
[tex]\[
\sqrt[3]{8x^3y^6} = \sqrt[3]{(2xy^2)^3}
\][/tex]
4. Apply the Cube Root:
The cube root of a perfect cube [tex]\((2xy^2)^3\)[/tex] is just the base itself:
[tex]\[
\sqrt[3]{(2xy^2)^3} = 2xy^2
\][/tex]
Thus, the simplified form of the given expression [tex]\(\sqrt[3]{4xy^5} \times \sqrt[3]{2x^2y}\)[/tex] is:
[tex]\[
\boxed{2xy^2}
\][/tex]
