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Answer :
Sure! Let's tackle the simplification of the expression step-by-step:
1. Original Expression:
- We start with the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex].
2. Rewrite Using Rational Exponents:
- Write the expression using rational exponents: [tex]\((567 x^9 y^{11})^{\frac{1}{4}}\)[/tex].
3. Factor and Separate the Expression:
- Break down the numeric part of 567 into factors. Notice that [tex]\(567 = 81 \cdot 7\)[/tex].
- So the expression becomes [tex]\((81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex].
4. Simplify Using Known Powers:
- Since 81 is a perfect fourth power, as [tex]\(81 = 3^4\)[/tex], simplify [tex]\((81)^{\frac{1}{4}}\)[/tex] to 3.
- The expression becomes [tex]\(3 \cdot (7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex].
5. Clarify the Exponents:
- Further express the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] by rewriting:
- [tex]\(x^{\frac{9}{4}} = x^{(2 + \frac{1}{4})} = x^{2.25}\)[/tex]
- [tex]\(y^{\frac{11}{4}} = y^{(2 + \frac{3}{4})} = y^{2.75}\)[/tex]
6. Final Simplified Expression:
- Put it all together to get the fully simplified expression:
- [tex]\(3 \cdot (7)^{\frac{1}{4}} \cdot x^{2.25} \cdot y^{2.75}\)[/tex].
This is the step-by-step simplification of the given expression using the properties of rational exponents.
1. Original Expression:
- We start with the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex].
2. Rewrite Using Rational Exponents:
- Write the expression using rational exponents: [tex]\((567 x^9 y^{11})^{\frac{1}{4}}\)[/tex].
3. Factor and Separate the Expression:
- Break down the numeric part of 567 into factors. Notice that [tex]\(567 = 81 \cdot 7\)[/tex].
- So the expression becomes [tex]\((81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex].
4. Simplify Using Known Powers:
- Since 81 is a perfect fourth power, as [tex]\(81 = 3^4\)[/tex], simplify [tex]\((81)^{\frac{1}{4}}\)[/tex] to 3.
- The expression becomes [tex]\(3 \cdot (7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}\)[/tex].
5. Clarify the Exponents:
- Further express the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] by rewriting:
- [tex]\(x^{\frac{9}{4}} = x^{(2 + \frac{1}{4})} = x^{2.25}\)[/tex]
- [tex]\(y^{\frac{11}{4}} = y^{(2 + \frac{3}{4})} = y^{2.75}\)[/tex]
6. Final Simplified Expression:
- Put it all together to get the fully simplified expression:
- [tex]\(3 \cdot (7)^{\frac{1}{4}} \cdot x^{2.25} \cdot y^{2.75}\)[/tex].
This is the step-by-step simplification of the given expression using the properties of rational exponents.
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