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Answer :
Sure, let's simplify [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] step-by-step.
### Step-by-Step Solution:
1. Write the Original Expressions:
We start with the expressions:
[tex]\[
\sqrt[3]{5x} \quad \text{and} \quad \sqrt[3]{25x^2}
\][/tex]
2. Convert to Exponential Form:
Both expressions are cube roots, so we can rewrite them using exponents:
[tex]\[
\sqrt[3]{5x} = (5x)^{1/3} \quad \text{and} \quad \sqrt[3]{25x^2} = (25x^2)^{1/3}
\][/tex]
3. Multiply the Expressions:
Now, we multiply these two expressions:
[tex]\[
(5x)^{1/3} \cdot (25x^2)^{1/3}
\][/tex]
4. Combine the Exponents:
When multiplying expressions with the same roots, we can combine them under a single root:
[tex]\[
(5x \cdot 25x^2)^{1/3}
\][/tex]
5. Simplify Inside the Parentheses:
Multiply the terms inside the parentheses:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
6. Simplify the Cube Root:
Take the cube root of the simplified term:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
Since [tex]\(125 = 5^3\)[/tex] and we have [tex]\(x^3\)[/tex], we get:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3 \cdot x^3} = 5x
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is:
[tex]\[
5x
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{5x}
\][/tex]
### Step-by-Step Solution:
1. Write the Original Expressions:
We start with the expressions:
[tex]\[
\sqrt[3]{5x} \quad \text{and} \quad \sqrt[3]{25x^2}
\][/tex]
2. Convert to Exponential Form:
Both expressions are cube roots, so we can rewrite them using exponents:
[tex]\[
\sqrt[3]{5x} = (5x)^{1/3} \quad \text{and} \quad \sqrt[3]{25x^2} = (25x^2)^{1/3}
\][/tex]
3. Multiply the Expressions:
Now, we multiply these two expressions:
[tex]\[
(5x)^{1/3} \cdot (25x^2)^{1/3}
\][/tex]
4. Combine the Exponents:
When multiplying expressions with the same roots, we can combine them under a single root:
[tex]\[
(5x \cdot 25x^2)^{1/3}
\][/tex]
5. Simplify Inside the Parentheses:
Multiply the terms inside the parentheses:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
6. Simplify the Cube Root:
Take the cube root of the simplified term:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
Since [tex]\(125 = 5^3\)[/tex] and we have [tex]\(x^3\)[/tex], we get:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3 \cdot x^3} = 5x
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is:
[tex]\[
5x
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{5x}
\][/tex]
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