Which expression is equivalent to [tex]\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}[/tex] ? Assume [tex]x \neq 0, y \neq 0[/tex].

A. [tex]\frac{y^8}{x^{10}}[/tex]
B. [tex]\frac{x^5}{y^7}[/tex]
C. [tex]\frac{x^5}{y^4}[/tex]
D. [tex]\frac{x}{y^7}[/tex]



Answer :

To determine the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex], we will simplify the expression step by step.

1. Simplify the base expression [tex]\(\frac{x^{-4} y}{x^{-9} y^5}\)[/tex]:

- The numerator is [tex]\(x^{-4} y\)[/tex].
- The denominator is [tex]\(x^{-9} y^5\)[/tex].

2. Combine the like bases in the fraction:

- For the [tex]\(x\)[/tex] terms, we use the rule of exponents [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex]:
[tex]\[ \frac{x^{-4}}{x^{-9}} = x^{-4 - (-9)} = x^{-4 + 9} = x^5 \][/tex]

- For the [tex]\(y\)[/tex] terms, we use the same exponent rule:
[tex]\[ \frac{y}{y^5} = y^{1-5} = y^{-4} \][/tex]

3. Combine the results:

[tex]\[ \frac{x^{-4} y}{x^{-9} y^5} = x^5 \cdot y^{-4} \][/tex]

4. Apply the outer exponent [tex]\(-2\)[/tex] to the simplified base:

- For [tex]\(x\)[/tex], we use the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (x^5)^{-2} = x^{5 \cdot -2} = x^{-10} \][/tex]

- For [tex]\(y\)[/tex], we use the same rule:
[tex]\[ (y^{-4})^{-2} = y^{-4 \cdot -2} = y^{8} \][/tex]

5. Combine the results of the power application:

[tex]\[ (x^5 \cdot y^{-4})^{-2} = x^{-10} \cdot y^8 \][/tex]

6. Express the result as a fraction:

[tex]\[ x^{-10} \cdot y^8 = \frac{y^8}{x^{10}} \][/tex]

Therefore, the expression equivalent to [tex]\(\left(\frac{x^{-4} y}{x^{-9} y^5}\right)^{-2}\)[/tex] is [tex]\(\frac{y^8}{x^{10}}\)[/tex].

Thus, the correct answer is [tex]\(\boxed{\frac{y^8}{x^{10}}}\)[/tex].