Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can use the properties of cube roots.
Here are the steps to simplify it:
1. Use the Property of Cube Roots:
The property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex] allows us to combine the terms under a single cube root. So, we have:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Cube Root:
Multiply the expressions inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
3. Simplify the Cube Root:
Recognize that [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex] and [tex]\(x^3\)[/tex] is a perfect cube:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} = 5x
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
The correct answer is [tex]\(\boxed{5x}\)[/tex].
Here are the steps to simplify it:
1. Use the Property of Cube Roots:
The property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex] allows us to combine the terms under a single cube root. So, we have:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Cube Root:
Multiply the expressions inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
3. Simplify the Cube Root:
Recognize that [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex] and [tex]\(x^3\)[/tex] is a perfect cube:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} = 5x
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
The correct answer is [tex]\(\boxed{5x}\)[/tex].
