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Sagot :
To simplify the given expression [tex]\(\frac{x y^{-6}}{x^{-4} y^2}\)[/tex] and eliminate the negative exponents, we can follow these steps:
1. Separate the numerator and the denominator:
- Numerator: [tex]\(x \cdot y^{-6}\)[/tex]
- Denominator: [tex]\(x^{-4} \cdot y^2\)[/tex]
2. Eliminate the negative exponents: To rewrite a term with a negative exponent, move it to the other part of the fraction, changing the sign of the exponent.
- [tex]\(y^{-6}\)[/tex] in the numerator can be moved to the denominator as [tex]\(y^6\)[/tex]
- [tex]\(x^{-4}\)[/tex] in the denominator can be moved to the numerator as [tex]\(x^4\)[/tex]
So, the expression becomes:
[tex]\[ \frac{x \cdot 1}{1 \cdot y^6} \cdot \frac{x^4}{y^2} \][/tex]
3. Combine the fractions:
[tex]\[ \frac{x \cdot x^4}{y^6 \cdot y^2} \][/tex]
4. Simplify by multiplying the terms:
[tex]\[ \frac{x^{1+4}}{y^{6+2}} = \frac{x^5}{y^8} \][/tex]
We cannot find a match directly in the provided choices. Let's simplify each given choice to see if any of them are equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex]:
1. [tex]\(\frac{x^4}{y^2 x^6 y^6}\)[/tex]:
[tex]\[ \frac{x^4}{x^6 y^2 y^6} = \frac{x^4}{x^6 y^{2+6}} = \frac{x^4}{x^6 y^8} = \frac{1}{x^{6-4} y^8} = \frac{1}{x^2 y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
2. [tex]\(\frac{x^4}{y^2 y^6}\)[/tex]:
[tex]\[ \frac{x^4}{y^{2+6}} = \frac{x^4}{y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
3. [tex]\(\frac{x^4}{y^2 x y^6}\)[/tex]:
[tex]\[ \frac{x^4}{y^2 x y^6} = \frac{x^4}{x y^{2+6}} = \frac{x^4}{x y^8} = \frac{x^{4-1}}{y^8} = \frac{x^3}{y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
4. [tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]:
[tex]\[ \frac{x^4 y^2}{x y^6} = \frac{x^4}{x} \cdot \frac{y^2}{y^6} = \frac{x^{4-1}}{y^{6-2}} = \frac{x^3}{y^4} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
None of the given choices are equivalent to the simplified expression [tex]\(\frac{x^5}{y^8}\)[/tex].
Considering the correct form obtained from the steps above, if we strictly have to pick a choice given, the one that aligns closest while understanding proper mathematical transformation could be reassessed. Nonetheless, based on mathematical principles, the simplified form is uniquely derived as explained. If there was an error in given options representation, the steps remain solid.
Note: If choices need reassessment by your problem setter, the essential derived form here is the correct simplified progression.
1. Separate the numerator and the denominator:
- Numerator: [tex]\(x \cdot y^{-6}\)[/tex]
- Denominator: [tex]\(x^{-4} \cdot y^2\)[/tex]
2. Eliminate the negative exponents: To rewrite a term with a negative exponent, move it to the other part of the fraction, changing the sign of the exponent.
- [tex]\(y^{-6}\)[/tex] in the numerator can be moved to the denominator as [tex]\(y^6\)[/tex]
- [tex]\(x^{-4}\)[/tex] in the denominator can be moved to the numerator as [tex]\(x^4\)[/tex]
So, the expression becomes:
[tex]\[ \frac{x \cdot 1}{1 \cdot y^6} \cdot \frac{x^4}{y^2} \][/tex]
3. Combine the fractions:
[tex]\[ \frac{x \cdot x^4}{y^6 \cdot y^2} \][/tex]
4. Simplify by multiplying the terms:
[tex]\[ \frac{x^{1+4}}{y^{6+2}} = \frac{x^5}{y^8} \][/tex]
We cannot find a match directly in the provided choices. Let's simplify each given choice to see if any of them are equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex]:
1. [tex]\(\frac{x^4}{y^2 x^6 y^6}\)[/tex]:
[tex]\[ \frac{x^4}{x^6 y^2 y^6} = \frac{x^4}{x^6 y^{2+6}} = \frac{x^4}{x^6 y^8} = \frac{1}{x^{6-4} y^8} = \frac{1}{x^2 y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
2. [tex]\(\frac{x^4}{y^2 y^6}\)[/tex]:
[tex]\[ \frac{x^4}{y^{2+6}} = \frac{x^4}{y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
3. [tex]\(\frac{x^4}{y^2 x y^6}\)[/tex]:
[tex]\[ \frac{x^4}{y^2 x y^6} = \frac{x^4}{x y^{2+6}} = \frac{x^4}{x y^8} = \frac{x^{4-1}}{y^8} = \frac{x^3}{y^8} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
4. [tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]:
[tex]\[ \frac{x^4 y^2}{x y^6} = \frac{x^4}{x} \cdot \frac{y^2}{y^6} = \frac{x^{4-1}}{y^{6-2}} = \frac{x^3}{y^4} \][/tex]
This is not equivalent to [tex]\(\frac{x^5}{y^8}\)[/tex].
None of the given choices are equivalent to the simplified expression [tex]\(\frac{x^5}{y^8}\)[/tex].
Considering the correct form obtained from the steps above, if we strictly have to pick a choice given, the one that aligns closest while understanding proper mathematical transformation could be reassessed. Nonetheless, based on mathematical principles, the simplified form is uniquely derived as explained. If there was an error in given options representation, the steps remain solid.
Note: If choices need reassessment by your problem setter, the essential derived form here is the correct simplified progression.
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