Answer :
We start with the expression
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the Cube Roots
Since multiplying cube roots is equivalent to taking the cube root of the product, we have
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2. Multiply the Expressions Inside the Radical
Multiply the numbers and the variable parts separately:
- For the numbers: [tex]$5 \times 25 = 125$[/tex].
- For the variables: [tex]$x \times x^2 = x^3$[/tex].
Thus, the expression inside the cube root simplifies to
[tex]$$
125x^3.
$$[/tex]
So now the expression becomes
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Take the Cube Root
Notice that [tex]$125$[/tex] is a perfect cube since
[tex]$$
125 = 5^3,
$$[/tex]
and also
[tex]$$
x^3 = (x)^3.
$$[/tex]
Taking the cube root of each factor separately:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x = 5x.
$$[/tex]
Thus, the simplified form is
[tex]$$
\boxed{5x}.
$$[/tex]
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}.
$$[/tex]
Step 1. Combine the Cube Roots
Since multiplying cube roots is equivalent to taking the cube root of the product, we have
[tex]$$
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}.
$$[/tex]
Step 2. Multiply the Expressions Inside the Radical
Multiply the numbers and the variable parts separately:
- For the numbers: [tex]$5 \times 25 = 125$[/tex].
- For the variables: [tex]$x \times x^2 = x^3$[/tex].
Thus, the expression inside the cube root simplifies to
[tex]$$
125x^3.
$$[/tex]
So now the expression becomes
[tex]$$
\sqrt[3]{125x^3}.
$$[/tex]
Step 3. Take the Cube Root
Notice that [tex]$125$[/tex] is a perfect cube since
[tex]$$
125 = 5^3,
$$[/tex]
and also
[tex]$$
x^3 = (x)^3.
$$[/tex]
Taking the cube root of each factor separately:
[tex]$$
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x = 5x.
$$[/tex]
Thus, the simplified form is
[tex]$$
\boxed{5x}.
$$[/tex]
