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Answer :
Sure! Let's simplify the given expression, [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Understanding the Expression:
We start with two cube roots multiplied together: [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex].
2. Combining the Radicals:
We can use the property of radicals that states [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
By using this property, we get:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the Terms Inside the Cube Root:
- Multiply the coefficients: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex].
So, we have:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplifying the Cube Root:
- Recognize that 125 is a perfect cube: [tex]\(125 = 5^3\)[/tex].
- Recognize that [tex]\(x^3\)[/tex] is also a perfect cube.
Therefore, [tex]\(\sqrt[3]{125x^3}\)[/tex] simplifies to:
[tex]\[
5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
1. Understanding the Expression:
We start with two cube roots multiplied together: [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex].
2. Combining the Radicals:
We can use the property of radicals that states [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
By using this property, we get:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the Terms Inside the Cube Root:
- Multiply the coefficients: [tex]\(5 \times 25 = 125\)[/tex].
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex].
So, we have:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplifying the Cube Root:
- Recognize that 125 is a perfect cube: [tex]\(125 = 5^3\)[/tex].
- Recognize that [tex]\(x^3\)[/tex] is also a perfect cube.
Therefore, [tex]\(\sqrt[3]{125x^3}\)[/tex] simplifies to:
[tex]\[
5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
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