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Answer :
Sure! Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Combine the cube roots:
Using the property of cube roots that says [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex], we can combine the expression as follows:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply the expressions under the cube root:
Multiply the terms inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
3. Simplify the cube root:
Now, take the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
- Recognize that [tex]\(125 = 5^3\)[/tex].
- The cube root of [tex]\(5^3\)[/tex] is 5.
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex].
Therefore:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
The fully simplified expression is [tex]\(5x\)[/tex].
So, the correct answer is: [tex]\(\boxed{5x}\)[/tex].
1. Combine the cube roots:
Using the property of cube roots that says [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex], we can combine the expression as follows:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply the expressions under the cube root:
Multiply the terms inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
3. Simplify the cube root:
Now, take the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
- Recognize that [tex]\(125 = 5^3\)[/tex].
- The cube root of [tex]\(5^3\)[/tex] is 5.
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex].
Therefore:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
The fully simplified expression is [tex]\(5x\)[/tex].
So, the correct answer is: [tex]\(\boxed{5x}\)[/tex].
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