To simplify the expression [tex]\(\sqrt[3]{56 x^7 y^5}\)[/tex], let's work through the steps systematically:
1. Factor 56 into a product including a perfect cube:
[tex]\[ 56 = 8 \times 7 \][/tex]
Note that [tex]\(8\)[/tex] is a perfect cube since [tex]\(8 = 2^3\)[/tex].
2. Factor [tex]\(x^7\)[/tex] into a product involving perfect cubes:
[tex]\[ x^7 = (x^6) \cdot x = (x^2)^3 \cdot x \][/tex]
Here, [tex]\( (x^2)^3 \)[/tex] is a perfect cube.
3. Factor [tex]\(y^5\)[/tex] into a product involving perfect cubes:
[tex]\[ y^5 = (y^3) \cdot y^2 = y \cdot (y^2)^3 \][/tex]
Similarly, [tex]\((y^2)^3\)[/tex] is a perfect cube.
4. Combine all terms under the cube root, factoring out perfect cubes:
[tex]\[
\sqrt[3]{56 x^7 y^5} = \sqrt[3]{8 \cdot 7 \cdot (x^2)^3 \cdot x \cdot y \cdot (y^2)^3}
\][/tex]
5. Extract the perfect cubes from under the cube root:
[tex]\[
= \sqrt[3]{2^3 \cdot 7 \cdot (x^2)^3 \cdot x \cdot y \cdot (y^2)^3}
= \sqrt[3]{2^3} \cdot \sqrt[3]{(x^2)^3} \cdot \sqrt[3]{(y^2)^3} \cdot \sqrt[3]{7xy}
= 2 \cdot x^2 \cdot y \cdot \sqrt[3]{7xy^2}
\][/tex]
So, the simplified form of [tex]\(\sqrt[3]{56 x^7 y^5}\)[/tex] is:
[tex]\[
2 x^2 y \sqrt[3]{7 x y^2}
\][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
