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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:
- The property of cube roots states that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]. This means we can multiply the quantities inside the cube roots directly.
2. Apply the property to the expression:
- So, [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(\sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
3. Calculate the product inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex].
- Hence, [tex]\(5x \cdot 25x^2 = 125x^3\)[/tex].
4. Simplify the cube root:
- Now we have [tex]\(\sqrt[3]{125x^3}\)[/tex].
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x)^3 = x^3\)[/tex].
5. Combine the results:
- Therefore, [tex]\(\sqrt[3]{125x^3} = 5x\)[/tex].
The simplified expression is [tex]\(5x\)[/tex].
So, the correct answer is:
[tex]\[5x\][/tex]
### Step-by-Step Solution:
1. Understand the property of cube roots:
- The property of cube roots states that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]. This means we can multiply the quantities inside the cube roots directly.
2. Apply the property to the expression:
- So, [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(\sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
3. Calculate the product inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex].
- Hence, [tex]\(5x \cdot 25x^2 = 125x^3\)[/tex].
4. Simplify the cube root:
- Now we have [tex]\(\sqrt[3]{125x^3}\)[/tex].
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x)^3 = x^3\)[/tex].
5. Combine the results:
- Therefore, [tex]\(\sqrt[3]{125x^3} = 5x\)[/tex].
The simplified expression is [tex]\(5x\)[/tex].
So, the correct answer is:
[tex]\[5x\][/tex]
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