Answer :
We start with the expression
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1: Use the property of cube roots that
[tex]$$
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}.
$$[/tex]
Thus, we combine the cube roots:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2: Multiply the radicands:
- Multiply the coefficients: [tex]$5 \cdot 25 = 125$[/tex].
- Multiply the variables: [tex]$x \cdot x^2 = x^3$[/tex].
This gives:
[tex]$$
(5x)(25x^2) = 125x^3.
$$[/tex]
So the expression becomes:
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3: Simplify the cube root by separating the numerical and variable parts:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}.
$$[/tex]
Since [tex]$125 = 5^3$[/tex], we have
[tex]$$
\sqrt[3]{125} = 5.
$$[/tex]
And for the variable,
[tex]$$
\sqrt[3]{x^3} = x.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
5 \cdot x = 5x.
$$[/tex]
Final Answer:
[tex]$$
5x.
$$[/tex]
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1: Use the property of cube roots that
[tex]$$
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}.
$$[/tex]
Thus, we combine the cube roots:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2: Multiply the radicands:
- Multiply the coefficients: [tex]$5 \cdot 25 = 125$[/tex].
- Multiply the variables: [tex]$x \cdot x^2 = x^3$[/tex].
This gives:
[tex]$$
(5x)(25x^2) = 125x^3.
$$[/tex]
So the expression becomes:
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3: Simplify the cube root by separating the numerical and variable parts:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}.
$$[/tex]
Since [tex]$125 = 5^3$[/tex], we have
[tex]$$
\sqrt[3]{125} = 5.
$$[/tex]
And for the variable,
[tex]$$
\sqrt[3]{x^3} = x.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
5 \cdot x = 5x.
$$[/tex]
Final Answer:
[tex]$$
5x.
$$[/tex]
