Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can use the property of cube roots which states:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Applying this property, we have:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
Now, let's multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
Calculating the coefficients and powers, we get:
[tex]\[
5 \cdot 25 = 125
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So the expression inside the cube root becomes:
[tex]\[
125x^3
\][/tex]
Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
We know that:
[tex]\[
\sqrt[3]{125} = 5 \quad \text{(since \(125 = 5^3\))}
\][/tex]
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
Therefore:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
The simplified expression is [tex]\(\boxed{5x}\)[/tex].
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Applying this property, we have:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
Now, let's multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
Calculating the coefficients and powers, we get:
[tex]\[
5 \cdot 25 = 125
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So the expression inside the cube root becomes:
[tex]\[
125x^3
\][/tex]
Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
We know that:
[tex]\[
\sqrt[3]{125} = 5 \quad \text{(since \(125 = 5^3\))}
\][/tex]
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
Therefore:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
The simplified expression is [tex]\(\boxed{5x}\)[/tex].
