Sure, let's solve the expression [tex]\(\sqrt[3]{125 x^3 y}\)[/tex] step by step:
1. Identify the components inside the cube root: The expression inside the cube root is [tex]\(125 x^3 y\)[/tex].
2. Break down the expression:
- The number [tex]\(125\)[/tex] is a constant.
- The variable [tex]\(x\)[/tex] is raised to the power of 3, which is [tex]\(x^3\)[/tex].
- The variable [tex]\(y\)[/tex] remains as [tex]\(y\)[/tex].
3. Take the cube root of each component separately:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x\)[/tex].
- The cube root of [tex]\(y\)[/tex] is [tex]\(y^{1/3}\)[/tex].
4. Combine the results:
- Combining the cube root of [tex]\(125\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(y\)[/tex], we get:
[tex]\[
\sqrt[3]{125 x^3 y} = 5 \cdot x \cdot y^{1/3}
\][/tex]
5. Express the result in a simplified form:
- We can write [tex]\(x y^{1/3}\)[/tex] as [tex]\((x^3 y)^{1/3}\)[/tex] because we are distributing the cube root over the product.
- Hence, we get:
[tex]\[
\sqrt[3]{125 x^3 y} = 5 (x^3 y)^{1/3}
\][/tex]
So, the simplified answer is:
[tex]\[
5 (x^3 y)^{1/3}
\][/tex]
This is the simplified form of the given cube root expression.
