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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], let's break it down into steps:
1. Understand cube roots: We are working with cube roots here, which means using the property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]. This allows us to combine the two cube roots into one.
2. Combine the expressions inside the cube roots:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the expressions inside the cube root:
- Multiply the constants: [tex]\(5 \cdot 25 = 125\)[/tex]
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex]
So, the expression becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the cube root:
- [tex]\(125\)[/tex] is a perfect cube, since [tex]\(5^3 = 125\)[/tex].
- [tex]\(x^3\)[/tex] is also a perfect cube, as it is written as [tex]\((x^1)^3\)[/tex].
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x\)[/tex].
Thus, the expression simplifies to:
[tex]\[
5x
\][/tex]
The correct answer is [tex]\(5x\)[/tex].
1. Understand cube roots: We are working with cube roots here, which means using the property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex]. This allows us to combine the two cube roots into one.
2. Combine the expressions inside the cube roots:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the expressions inside the cube root:
- Multiply the constants: [tex]\(5 \cdot 25 = 125\)[/tex]
- Multiply the variables: [tex]\(x \cdot x^2 = x^3\)[/tex]
So, the expression becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the cube root:
- [tex]\(125\)[/tex] is a perfect cube, since [tex]\(5^3 = 125\)[/tex].
- [tex]\(x^3\)[/tex] is also a perfect cube, as it is written as [tex]\((x^1)^3\)[/tex].
Therefore, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x\)[/tex].
Thus, the expression simplifies to:
[tex]\[
5x
\][/tex]
The correct answer is [tex]\(5x\)[/tex].
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