College

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

A. [tex]$25 x^3$[/tex]

B. [tex][tex]$25 x$[/tex][/tex]

C. [tex]$5 x^3$[/tex]

D. [tex]$5 x$[/tex]

Answer :

To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can follow these steps:

1. Combine the Cube Roots:

The expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] can be combined under a single cube root because [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].

[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]

2. Multiply Inside the Cube Root:

Compute the product inside the cube root:

[tex]\[
(5x) \cdot (25x^2) = 5 \times 25 \times x \times x^2 = 125x^3
\][/tex]

3. Simplify the Cube Root:

The expression [tex]\(\sqrt[3]{125x^3}\)[/tex] can be simplified. First, recognize that 125 is a perfect cube: [tex]\(125 = 5^3\)[/tex].

Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3}\)[/tex].

Using the property of cube roots, [tex]\(\sqrt[3]{a^3} = a\)[/tex], we have:

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

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

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