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Simplify [tex]$\sqrt[3]{5 x} \cdot \sqrt[3]{25 x^2}$[/tex] completely.

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

Answer :

Sure, let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] step by step.

First, we recall a key property of cube roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]

Using this property, we can combine the two cube roots into a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]

Next, we perform the multiplication inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
[tex]\[
5 \cdot 25 = 125
\][/tex]
[tex]\[
x \cdot x^2 = x^3
\][/tex]

Thus, the expression inside the cube root becomes:
[tex]\[
(5x) \cdot (25x^2) = 125x^3
\][/tex]

Now we have:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]

We know that [tex]\(\sqrt[3]{a^3} = a\)[/tex]. Therefore:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
[tex]\[
\sqrt[3]{125} = 5 \quad \text{(since } 125 = 5^3\text{)}
\][/tex]
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]

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

Therefore, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is:
[tex]\[
5x
\][/tex]

The correct answer is:
[tex]\[
5x
\][/tex]

Thanks for taking the time to read Simplify tex sqrt 3 5 x cdot sqrt 3 25 x 2 tex completely A tex 25 x 3 tex B tex 25 x tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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