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Answer :
To simplify the radical expression [tex]\(\sqrt[4]{625 x^{12} y^8}\)[/tex], follow these steps:
1. Simplify the fourth root of 625:
- Recognize that 625 is a perfect fourth power. Specifically, [tex]\(625 = 5^4\)[/tex].
- Therefore, the fourth root of 625 is [tex]\(5\)[/tex].
2. Simplify the fourth root of [tex]\(x^{12}\)[/tex]:
- Apply the rule for taking roots of powers: [tex]\(\sqrt[4]{x^{12}} = x^{12/4}\)[/tex].
- Simplify the exponent: [tex]\(12/4 = 3\)[/tex].
- So, [tex]\(\sqrt[4]{x^{12}} = x^3\)[/tex].
3. Simplify the fourth root of [tex]\(y^8\)[/tex]:
- Use the same rule for taking roots of powers: [tex]\(\sqrt[4]{y^8} = y^{8/4}\)[/tex].
- Simplify the exponent: [tex]\(8/4 = 2\)[/tex].
- Thus, [tex]\(\sqrt[4]{y^8} = y^2\)[/tex].
4. Combine the results:
- By putting it all together, we combine [tex]\(5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(y^2\)[/tex].
- Since we are taking even roots, we use absolute value notation for the parts that could be negative if variables represent negative numbers.
- Thus, the simplified expression is [tex]\(5 |x^3| y^2\)[/tex].
So, the correct option is [tex]\(5 |x^3| y^2\)[/tex].
1. Simplify the fourth root of 625:
- Recognize that 625 is a perfect fourth power. Specifically, [tex]\(625 = 5^4\)[/tex].
- Therefore, the fourth root of 625 is [tex]\(5\)[/tex].
2. Simplify the fourth root of [tex]\(x^{12}\)[/tex]:
- Apply the rule for taking roots of powers: [tex]\(\sqrt[4]{x^{12}} = x^{12/4}\)[/tex].
- Simplify the exponent: [tex]\(12/4 = 3\)[/tex].
- So, [tex]\(\sqrt[4]{x^{12}} = x^3\)[/tex].
3. Simplify the fourth root of [tex]\(y^8\)[/tex]:
- Use the same rule for taking roots of powers: [tex]\(\sqrt[4]{y^8} = y^{8/4}\)[/tex].
- Simplify the exponent: [tex]\(8/4 = 2\)[/tex].
- Thus, [tex]\(\sqrt[4]{y^8} = y^2\)[/tex].
4. Combine the results:
- By putting it all together, we combine [tex]\(5\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(y^2\)[/tex].
- Since we are taking even roots, we use absolute value notation for the parts that could be negative if variables represent negative numbers.
- Thus, the simplified expression is [tex]\(5 |x^3| y^2\)[/tex].
So, the correct option is [tex]\(5 |x^3| y^2\)[/tex].
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