College

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 :

We start with the expression

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

Step 1. Combine the cube roots

Since the cube root of a product is the product of the cube roots, we combine them under a single cube root:

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

Step 2. Multiply the expressions inside the cube root

Multiply the coefficients and the variables separately:

- Coefficients: [tex]$5 \cdot 25 = 125$[/tex].
- Variables: [tex]$x \cdot x^2 = x^3$[/tex].

Thus, the inside of the cube root becomes:

[tex]$$
125x^3.
$$[/tex]

So now we have:

[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]

Step 3. Simplify the cube root

Recognize that [tex]$125 = 5^3$[/tex], and that the cube root of [tex]$x^3$[/tex] is [tex]$x$[/tex]. Thus,

[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} = 5 \cdot x.
$$[/tex]

Final Answer

The simplified expression is

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