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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 :

We start with the expression

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

Step 1. Combine the Cube Roots

Since multiplying cube roots is equivalent to taking the cube root of the product, we have

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

Step 2. Multiply the Expressions Inside the Radical

Multiply the numbers and the variable parts separately:

- For the numbers: [tex]$5 \times 25 = 125$[/tex].
- For the variables: [tex]$x \times x^2 = x^3$[/tex].

Thus, the expression inside the cube root simplifies to

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

So now the expression becomes

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

Step 3. Take the Cube Root

Notice that [tex]$125$[/tex] is a perfect cube since

[tex]$$
125 = 5^3,
$$[/tex]

and also

[tex]$$
x^3 = (x)^3.
$$[/tex]

Taking the cube root of each factor separately:

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

Thus, the simplified form is

[tex]$$
\boxed{5x}.
$$[/tex]

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