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Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can follow a step-by-step approach:
1. Understand Cube Roots: Cube roots work similarly to square roots but deal with powers of three. The cube root [tex]\(\sqrt[3]{a}\)[/tex] is equivalent to [tex]\(a^{1/3}\)[/tex].
2. Apply Cube Root Properties: We know that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
3. Multiply Inside the Cube Root: Using the property above, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(\sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
4. Multiply the Terms: Inside the cube root, multiply the coefficients and the variables:
- Coefficients: [tex]\(5 \cdot 25 = 125\)[/tex]
- Variables: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex]
5. Simplify Inside the Cube Root: Now the expression inside the cube root is [tex]\(125x^3\)[/tex].
6. Take the Cube Root:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
7. Combine the Results: The expression simplifies to [tex]\(5x\)[/tex].
Therefore, the completely simplified expression is [tex]\(5x\)[/tex].
1. Understand Cube Roots: Cube roots work similarly to square roots but deal with powers of three. The cube root [tex]\(\sqrt[3]{a}\)[/tex] is equivalent to [tex]\(a^{1/3}\)[/tex].
2. Apply Cube Root Properties: We know that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
3. Multiply Inside the Cube Root: Using the property above, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] becomes [tex]\(\sqrt[3]{(5x) \cdot (25x^2)}\)[/tex].
4. Multiply the Terms: Inside the cube root, multiply the coefficients and the variables:
- Coefficients: [tex]\(5 \cdot 25 = 125\)[/tex]
- Variables: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex]
5. Simplify Inside the Cube Root: Now the expression inside the cube root is [tex]\(125x^3\)[/tex].
6. Take the Cube Root:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
7. Combine the Results: The expression simplifies to [tex]\(5x\)[/tex].
Therefore, the completely simplified expression is [tex]\(5x\)[/tex].
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