Answer :
We start with the expression
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the cube roots into a single cube root by multiplying under the radical:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}.
$$[/tex]
Step 2. Multiply the radicands:
- Multiply the numerical parts: [tex]$5 \cdot 25 = 125$[/tex].
- Multiply the variable parts: [tex]$x \cdot x^2 = x^3$[/tex].
Thus, the radicand becomes:
[tex]$$
125x^3.
$$[/tex]
So we have
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Rewrite the cube root as the product of cube roots of the individual factors:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}.
$$[/tex]
Step 4. Simplify each cube root separately:
- Since [tex]$125 = 5^3$[/tex], we have
[tex]$$
\sqrt[3]{125} = 5.
$$[/tex]
- For the variable [tex]$x^3$[/tex], we know that
[tex]$$
\sqrt[3]{x^3} = x.
$$[/tex]
Step 5. Combine the results:
[tex]$$
\sqrt[3]{125x^3} = 5 \cdot x = 5x.
$$[/tex]
Thus, the simplified form of the expression is
[tex]$$
\boxed{5x}.
$$[/tex]
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the cube roots into a single cube root by multiplying under the radical:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}.
$$[/tex]
Step 2. Multiply the radicands:
- Multiply the numerical parts: [tex]$5 \cdot 25 = 125$[/tex].
- Multiply the variable parts: [tex]$x \cdot x^2 = x^3$[/tex].
Thus, the radicand becomes:
[tex]$$
125x^3.
$$[/tex]
So we have
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Rewrite the cube root as the product of cube roots of the individual factors:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}.
$$[/tex]
Step 4. Simplify each cube root separately:
- Since [tex]$125 = 5^3$[/tex], we have
[tex]$$
\sqrt[3]{125} = 5.
$$[/tex]
- For the variable [tex]$x^3$[/tex], we know that
[tex]$$
\sqrt[3]{x^3} = x.
$$[/tex]
Step 5. Combine the results:
[tex]$$
\sqrt[3]{125x^3} = 5 \cdot x = 5x.
$$[/tex]
Thus, the simplified form of the expression is
[tex]$$
\boxed{5x}.
$$[/tex]
