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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], let's go through the steps:
1. Identify the cube roots:
- The first term is [tex]\(\sqrt[3]{5x}\)[/tex], which can be expressed as [tex]\((5x)^{1/3}\)[/tex].
- The second term is [tex]\(\sqrt[3]{25x^2}\)[/tex], which can be expressed as [tex]\((25x^2)^{1/3}\)[/tex].
2. Combine under a single cube root:
- We can multiply these two expressions under the same cube root:
[tex]\(((5x)^{1/3}) \times ((25x^2)^{1/3})\)[/tex].
3. Simplify under the cube root using the properties of exponents:
- When multiplying with the same root, we can combine the bases and exponents:
[tex]\((5x \cdot 25x^2)^{1/3}\)[/tex].
4. Multiply the expressions inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex].
5. Substitute back into the cube root:
- [tex]\( (125x^3)^{1/3} \)[/tex].
6. Simplify the cube root:
- Recognize that [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex] and [tex]\(x^3\)[/tex] is also already a perfect cube of [tex]\(x\)[/tex].
- [tex]\((125x^3)^{1/3} = (5^3 \cdot x^3)^{1/3} = 5x\)[/tex].
So, the simplified expression is [tex]\(5x\)[/tex].
The correct simplified result from the provided list of options is [tex]\(5x\)[/tex].
1. Identify the cube roots:
- The first term is [tex]\(\sqrt[3]{5x}\)[/tex], which can be expressed as [tex]\((5x)^{1/3}\)[/tex].
- The second term is [tex]\(\sqrt[3]{25x^2}\)[/tex], which can be expressed as [tex]\((25x^2)^{1/3}\)[/tex].
2. Combine under a single cube root:
- We can multiply these two expressions under the same cube root:
[tex]\(((5x)^{1/3}) \times ((25x^2)^{1/3})\)[/tex].
3. Simplify under the cube root using the properties of exponents:
- When multiplying with the same root, we can combine the bases and exponents:
[tex]\((5x \cdot 25x^2)^{1/3}\)[/tex].
4. Multiply the expressions inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex].
5. Substitute back into the cube root:
- [tex]\( (125x^3)^{1/3} \)[/tex].
6. Simplify the cube root:
- Recognize that [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex] and [tex]\(x^3\)[/tex] is also already a perfect cube of [tex]\(x\)[/tex].
- [tex]\((125x^3)^{1/3} = (5^3 \cdot x^3)^{1/3} = 5x\)[/tex].
So, the simplified expression is [tex]\(5x\)[/tex].
The correct simplified result from the provided list of options is [tex]\(5x\)[/tex].
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