Answer :
Sure! Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
### Step-by-step Solution:
1. Understand the Property of Cube Roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the cube roots.
2. Combine the Expressions:
- Here, [tex]\(a = 5x\)[/tex] and [tex]\(b = 25x^2\)[/tex]. So:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply Inside the Cube Root:
- Multiply the coefficients and the variables separately.
- Coefficients: [tex]\(5 \cdot 25 = 125\)[/tex]
- Variables: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex]
Therefore, the expression becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the Cube Root:
- Break it down and simplify:
[tex]\[
\sqrt[3]{125} = 5 \quad \text{(because \(5^3 = 125\))}
\][/tex]
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
### Step-by-step Solution:
1. Understand the Property of Cube Roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the cube roots.
2. Combine the Expressions:
- Here, [tex]\(a = 5x\)[/tex] and [tex]\(b = 25x^2\)[/tex]. So:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply Inside the Cube Root:
- Multiply the coefficients and the variables separately.
- Coefficients: [tex]\(5 \cdot 25 = 125\)[/tex]
- Variables: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex]
Therefore, the expression becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the Cube Root:
- Break it down and simplify:
[tex]\[
\sqrt[3]{125} = 5 \quad \text{(because \(5^3 = 125\))}
\][/tex]
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
