Answer :
We start with the expression
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Since the cube root of a product is the product of the cube roots, we can combine the radicands:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Multiply the coefficients and add the exponents of [tex]$x$[/tex]:
[tex]$$
(5x)(25x^2) = 125x^3.
$$[/tex]
Now, we have
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Recognize that [tex]$125 = 5^3$[/tex] and [tex]$x^3$[/tex] is a perfect cube. Taking the cube root of each part gives:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x.
$$[/tex]
Thus, the simplified expression is
[tex]$$
5x.
$$[/tex]
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Since the cube root of a product is the product of the cube roots, we can combine the radicands:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Multiply the coefficients and add the exponents of [tex]$x$[/tex]:
[tex]$$
(5x)(25x^2) = 125x^3.
$$[/tex]
Now, we have
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Recognize that [tex]$125 = 5^3$[/tex] and [tex]$x^3$[/tex] is a perfect cube. Taking the cube root of each part gives:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x.
$$[/tex]
Thus, the simplified expression is
[tex]$$
5x.
$$[/tex]
