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Choose the simplified form of [tex]\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}[/tex] completely.

A. [tex]25x^3[/tex]
B. [tex]25x[/tex]
C. [tex]5x^3[/tex]
D. [tex]5x[/tex]

Answer :

Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].

Here is a step-by-step solution:

1. Combine the Cube Roots:
When you multiply cube roots, you can combine them under one cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]

2. Multiply Inside the Cube Root:
Multiply the terms inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]

3. Simplify the Cube Root:
Now, you need to simplify [tex]\(\sqrt[3]{125x^3}\)[/tex].

- Cube Root of 125: Since [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex], the cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex].
- Cube Root of [tex]\(x^3\)[/tex]: The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex].

Therefore:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]

So, the expression simplifies to [tex]\(5x\)[/tex].

The correct answer is [tex]\(\boxed{5x}\)[/tex].

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