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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can follow these steps:
1. Combine the Cube Roots:
The expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] can be combined under a single cube root because [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Cube Root:
Compute the product inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \times 25 \times x \times x^2 = 125x^3
\][/tex]
3. Simplify the Cube Root:
The expression [tex]\(\sqrt[3]{125x^3}\)[/tex] can be simplified. First, recognize that 125 is a perfect cube: [tex]\(125 = 5^3\)[/tex].
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3}\)[/tex].
Using the property of cube roots, [tex]\(\sqrt[3]{a^3} = a\)[/tex], we have:
[tex]\[
\sqrt[3]{5^3x^3} = 5x
\][/tex]
Thus, the simplified expression is [tex]\(5x\)[/tex].
The correct answer is: [tex]\(5x\)[/tex].
1. Combine the Cube Roots:
The expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] can be combined under a single cube root because [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Cube Root:
Compute the product inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \times 25 \times x \times x^2 = 125x^3
\][/tex]
3. Simplify the Cube Root:
The expression [tex]\(\sqrt[3]{125x^3}\)[/tex] can be simplified. First, recognize that 125 is a perfect cube: [tex]\(125 = 5^3\)[/tex].
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3}\)[/tex].
Using the property of cube roots, [tex]\(\sqrt[3]{a^3} = a\)[/tex], we have:
[tex]\[
\sqrt[3]{5^3x^3} = 5x
\][/tex]
Thus, the simplified expression is [tex]\(5x\)[/tex].
The correct answer is: [tex]\(5x\)[/tex].
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