To solve the problem of finding the expression equivalent to [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can use the product property of roots. The product property of roots states that for any positive numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] and any integer [tex]\(n\)[/tex]:
[tex]\[
\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}
\][/tex]
In this case, since both roots are cube roots ([tex]\(n = 3\)[/tex]), we can combine them under a single cube root. Let's start with the given expressions:
[tex]\[
\sqrt[3]{5x} \text{ and } \sqrt[3]{25x^2}
\][/tex]
Using the product property:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
Next, we multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
Calculate the numerical multiplication:
[tex]\[
5 \cdot 25 = 125
\][/tex]
And combine the powers of [tex]\(x\)[/tex]:
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So, we have:
[tex]\[
(5x) \cdot (25x^2) = 125x^3
\][/tex]
Thus, combining these within the cube root gives us:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
Therefore, the expression equivalent to [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is:
[tex]\[
\boxed{\sqrt[3]{125x^3}}
\][/tex]
