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Simplify [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 :

To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can use the property of cube roots: [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \times b}\)[/tex]. This lets us combine the two cube roots into a single cube root:

1. Multiply the expressions inside the cube roots:
[tex]\[ 5x \times 25x^2 = 125x^3 \][/tex]

2. Now, take the cube root of the product:
[tex]\[ \sqrt[3]{125x^3} \][/tex]

3. Simplify the cube root:
- [tex]\(\sqrt[3]{125}\)[/tex] simplifies to 5 because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
- [tex]\(\sqrt[3]{x^3}\)[/tex] simplifies to [tex]\(x\)[/tex] because the cube root and the exponent cancel each other out.

4. Combine these results:
[tex]\[ \sqrt[3]{125x^3} = 5x \][/tex]

So, the completely simplified expression is [tex]\(5x\)[/tex].

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