Answer :
We start with the expression
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the cube roots
Since the cube root of a product is the product of the cube roots, we combine them under a single cube root:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2. Multiply the expressions inside the cube root
Multiply the coefficients and the variables separately:
- Coefficients: [tex]$5 \cdot 25 = 125$[/tex].
- Variables: [tex]$x \cdot x^2 = x^3$[/tex].
Thus, the inside of the cube root becomes:
[tex]$$
125x^3.
$$[/tex]
So now we have:
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Simplify the cube root
Recognize that [tex]$125 = 5^3$[/tex], and that the cube root of [tex]$x^3$[/tex] is [tex]$x$[/tex]. Thus,
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} = 5 \cdot x.
$$[/tex]
Final Answer
The simplified expression is
[tex]$$
5x.
$$[/tex]
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the cube roots
Since the cube root of a product is the product of the cube roots, we combine them under a single cube root:
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2. Multiply the expressions inside the cube root
Multiply the coefficients and the variables separately:
- Coefficients: [tex]$5 \cdot 25 = 125$[/tex].
- Variables: [tex]$x \cdot x^2 = x^3$[/tex].
Thus, the inside of the cube root becomes:
[tex]$$
125x^3.
$$[/tex]
So now we have:
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Simplify the cube root
Recognize that [tex]$125 = 5^3$[/tex], and that the cube root of [tex]$x^3$[/tex] is [tex]$x$[/tex]. Thus,
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3} = \sqrt[3]{5^3} \cdot \sqrt[3]{x^3} = 5 \cdot x.
$$[/tex]
Final Answer
The simplified expression is
[tex]$$
5x.
$$[/tex]
