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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we'll use the property of cube roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
1. Identify the Terms:
- We have [tex]\(a = 5x\)[/tex] and [tex]\(b = 25x^2\)[/tex].
2. Multiply the Terms Inside the Cube Root:
- First, multiply the numerical coefficients: [tex]\(5 \cdot 25 = 125\)[/tex].
- Next, multiply the [tex]\(x\)[/tex] terms: [tex]\(x \cdot x^2 = x^3\)[/tex].
3. Combine the Results:
- This gives us: [tex]\(\sqrt[3]{125x^3}\)[/tex].
4. Simplify the Cube Root:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] ([tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex]).
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] (since [tex]\((x) \cdot (x) \cdot (x) = x^3\)[/tex]).
5. Combine the Simplified Parts:
- Thus, [tex]\(\sqrt[3]{125x^3} = 5x\)[/tex].
So, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(\boxed{5x}\)[/tex].
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
1. Identify the Terms:
- We have [tex]\(a = 5x\)[/tex] and [tex]\(b = 25x^2\)[/tex].
2. Multiply the Terms Inside the Cube Root:
- First, multiply the numerical coefficients: [tex]\(5 \cdot 25 = 125\)[/tex].
- Next, multiply the [tex]\(x\)[/tex] terms: [tex]\(x \cdot x^2 = x^3\)[/tex].
3. Combine the Results:
- This gives us: [tex]\(\sqrt[3]{125x^3}\)[/tex].
4. Simplify the Cube Root:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] ([tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex]).
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] (since [tex]\((x) \cdot (x) \cdot (x) = x^3\)[/tex]).
5. Combine the Simplified Parts:
- Thus, [tex]\(\sqrt[3]{125x^3} = 5x\)[/tex].
So, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(\boxed{5x}\)[/tex].
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