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Answer :
Sure! Let's go through the problem step-by-step.
We need to simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Understand the Cube Root Property:
The product of cube roots can be combined as a single cube root:
[tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
So, [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
2. Multiply Inside the Cube Root:
Now, multiply the expressions inside the cube root:
[tex]\((5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2\)[/tex].
3. Simplify the Multiplication:
- Multiply the coefficients: [tex]\(5 \cdot 25 = 125\)[/tex].
- Combine the powers of [tex]\(x\)[/tex]: [tex]\(x \cdot x^2 = x^3\)[/tex].
So, the expression inside the cube root becomes [tex]\(125x^3\)[/tex].
4. Simplify the Cube Root:
We need to find [tex]\(\sqrt[3]{125x^3}\)[/tex].
- The cube root of 125 is 5 because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Combine the Results:
Therefore, [tex]\(\sqrt[3]{125x^3}\)[/tex] simplifies to [tex]\(5x\)[/tex].
So, the final simplified result of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(\boxed{5x}\)[/tex].
We need to simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Understand the Cube Root Property:
The product of cube roots can be combined as a single cube root:
[tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
So, [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
2. Multiply Inside the Cube Root:
Now, multiply the expressions inside the cube root:
[tex]\((5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2\)[/tex].
3. Simplify the Multiplication:
- Multiply the coefficients: [tex]\(5 \cdot 25 = 125\)[/tex].
- Combine the powers of [tex]\(x\)[/tex]: [tex]\(x \cdot x^2 = x^3\)[/tex].
So, the expression inside the cube root becomes [tex]\(125x^3\)[/tex].
4. Simplify the Cube Root:
We need to find [tex]\(\sqrt[3]{125x^3}\)[/tex].
- The cube root of 125 is 5 because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Combine the Results:
Therefore, [tex]\(\sqrt[3]{125x^3}\)[/tex] simplifies to [tex]\(5x\)[/tex].
So, the final simplified result of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(\boxed{5x}\)[/tex].
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