Answer :
Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Understand the Expression:
We're given two cube roots:
- [tex]\(\sqrt[3]{5x}\)[/tex]
- [tex]\(\sqrt[3]{25x^2}\)[/tex]
2. Use the Property of Cube Roots:
The property of cube roots allows us to combine them:
[tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]
So, we have:
[tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}\)[/tex]
3. Multiply Inside the Cube Root:
Multiply the terms inside the cube root:
[tex]\((5x) \cdot (25x^2) = 125x^3\)[/tex]
4. Simplify the Cube Root:
[tex]\(\sqrt[3]{125x^3}\)[/tex] is what we get next.
Notice that [tex]\(125\)[/tex] and [tex]\(x^3\)[/tex] are both perfect cubes:
- [tex]\(125 = 5^3\)[/tex]
- [tex]\(x^3\)[/tex] is already a cube
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{(5x)^3}\)[/tex]
5. Evaluate the Cube Root of a Perfect Cube:
The cube root of a perfect cube like [tex]\((5x)^3\)[/tex] is simply [tex]\(5x\)[/tex].
So, after simplifying, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
1. Understand the Expression:
We're given two cube roots:
- [tex]\(\sqrt[3]{5x}\)[/tex]
- [tex]\(\sqrt[3]{25x^2}\)[/tex]
2. Use the Property of Cube Roots:
The property of cube roots allows us to combine them:
[tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]
So, we have:
[tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}\)[/tex]
3. Multiply Inside the Cube Root:
Multiply the terms inside the cube root:
[tex]\((5x) \cdot (25x^2) = 125x^3\)[/tex]
4. Simplify the Cube Root:
[tex]\(\sqrt[3]{125x^3}\)[/tex] is what we get next.
Notice that [tex]\(125\)[/tex] and [tex]\(x^3\)[/tex] are both perfect cubes:
- [tex]\(125 = 5^3\)[/tex]
- [tex]\(x^3\)[/tex] is already a cube
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{(5x)^3}\)[/tex]
5. Evaluate the Cube Root of a Perfect Cube:
The cube root of a perfect cube like [tex]\((5x)^3\)[/tex] is simply [tex]\(5x\)[/tex].
So, after simplifying, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
