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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], let's break it down:
1. Combine the Radicals:
- We have two cube roots being multiplied: [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex]. According to the properties of radicals, when you multiply them, you can combine them under a single radical:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Radical:
- Now, 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, simplify the cube root of [tex]\(125x^3\)[/tex]:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\(x^3\)[/tex] is a perfect cube, [tex]\((x \cdot x \cdot x)\)[/tex].
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] simplifies to [tex]\(\boxed{5x}\)[/tex].
1. Combine the Radicals:
- We have two cube roots being multiplied: [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex]. According to the properties of radicals, when you multiply them, you can combine them under a single radical:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply Inside the Radical:
- Now, 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, simplify the cube root of [tex]\(125x^3\)[/tex]:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\(x^3\)[/tex] is a perfect cube, [tex]\((x \cdot x \cdot x)\)[/tex].
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] simplifies to [tex]\(\boxed{5x}\)[/tex].
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