Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can use the properties of exponents and roots. Let's go through the process step-by-step:
1. Express Each Radical in Exponent Form:
- The cube root of a number can be written with an exponent of [tex]\(\frac{1}{3}\)[/tex]. Therefore, [tex]\(\sqrt[3]{5x}\)[/tex] is expressed as [tex]\((5x)^{1/3}\)[/tex], and [tex]\(\sqrt[3]{25x^2}\)[/tex] is expressed as [tex]\((25x^2)^{1/3}\)[/tex].
2. Multiply the Expressions:
- When multiplying expressions with the same root, you can combine them under a single radical. Thus, [tex]\((5x)^{1/3} \cdot (25x^2)^{1/3}\)[/tex] becomes [tex]\((5x \cdot 25x^2)^{1/3}\)[/tex].
3. Simplify Inside the Parentheses:
- Multiply the constants: [tex]\(5 \times 25 = 125\)[/tex].
- Combine the powers of [tex]\(x\)[/tex]: [tex]\(x \times x^2 = x^3\)[/tex].
So, the expression inside the parentheses simplifies to [tex]\(125x^3\)[/tex].
4. Find the Cube Root:
- Now we need to find the cube root of [tex]\(125x^3\)[/tex]. 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].
5. Final Simplified Expression:
- Combine the results: [tex]\(5x\)[/tex].
The completely simplified result of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex]. This means the correct choice out of the given options is [tex]\(5x\)[/tex].
1. Express Each Radical in Exponent Form:
- The cube root of a number can be written with an exponent of [tex]\(\frac{1}{3}\)[/tex]. Therefore, [tex]\(\sqrt[3]{5x}\)[/tex] is expressed as [tex]\((5x)^{1/3}\)[/tex], and [tex]\(\sqrt[3]{25x^2}\)[/tex] is expressed as [tex]\((25x^2)^{1/3}\)[/tex].
2. Multiply the Expressions:
- When multiplying expressions with the same root, you can combine them under a single radical. Thus, [tex]\((5x)^{1/3} \cdot (25x^2)^{1/3}\)[/tex] becomes [tex]\((5x \cdot 25x^2)^{1/3}\)[/tex].
3. Simplify Inside the Parentheses:
- Multiply the constants: [tex]\(5 \times 25 = 125\)[/tex].
- Combine the powers of [tex]\(x\)[/tex]: [tex]\(x \times x^2 = x^3\)[/tex].
So, the expression inside the parentheses simplifies to [tex]\(125x^3\)[/tex].
4. Find the Cube Root:
- Now we need to find the cube root of [tex]\(125x^3\)[/tex]. 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].
5. Final Simplified Expression:
- Combine the results: [tex]\(5x\)[/tex].
The completely simplified result of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex]. This means the correct choice out of the given options is [tex]\(5x\)[/tex].
