Certainly! Let's break down the solution for the expression [tex]\(\sqrt[3]{64 x^3 y^7}\)[/tex] step by step:
1. Understand the given expression: We start with the expression [tex]\(\sqrt[3]{64 x^3 y^7}\)[/tex]. This is a cube root of a product involving constant [tex]\(64\)[/tex] and variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
2. Prime factorization inside the cube root:
- The constant [tex]\(64\)[/tex] can be written as [tex]\(64 = 4^3\)[/tex], because [tex]\(64 = 4 \times 4 \times 4\)[/tex].
3. Rewrite the expression using factorization:
[tex]\[
\sqrt[3]{64 x^3 y^7} = \sqrt[3]{4^3 \cdot x^3 \cdot y^7}
\][/tex]
4. Distribute the cube root over the product: The cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{4^3 \cdot x^3 \cdot y^7} = \sqrt[3]{4^3} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y^7}
\][/tex]
5. Simplify each cube root individually:
- [tex]\(\sqrt[3]{4^3}\)[/tex] is simply [tex]\(4\)[/tex], because the cube root and the cube cancel each other.
- [tex]\(\sqrt[3]{x^3}\)[/tex] is [tex]\(x\)[/tex], for the same reason.
- [tex]\(\sqrt[3]{y^7}\)[/tex] requires a bit more work. Notice that [tex]\(y^7 = y^6 \cdot y\)[/tex]. Therefore,
[tex]\[
\sqrt[3]{y^7} = \sqrt[3]{(y^6 \cdot y)} = \sqrt[3]{y^6} \cdot \sqrt[3]{y} = y^2 \cdot y^{1/3}
\][/tex]
because [tex]\(\sqrt[3]{y^6} = (y^6)^{1/3} = y^{6/3} = y^2\)[/tex].
6. Combine all parts together:
[tex]\[
\sqrt[3]{64 x^3 y^7} = 4 \cdot x \cdot y^2 \cdot y^{1/3}
\][/tex]
7. Express the final result:
[tex]\[
\sqrt[3]{64 x^3 y^7} = 4 x y^2 y^{1/3} = 4 \cdot (x^3 y^7)^{1/3}
\][/tex]
Thus, after simplifying, the given expression [tex]\(\sqrt[3]{64 x^3 y^7}\)[/tex] evaluates to:
[tex]\[
4 \cdot (x^3 y^7)^{1/3}
\][/tex]
