To simplify the expression \((3x - 4y^5)^{-2}\), we can rewrite this as:
\[ \frac{1}{(3x - 4y^5)^2} \]
Now, we need to expand \((3x - 4y^5)^2\):
\[ (3x - 4y^5)^2 = (3x)^2 - 2(3x)(4y^5) + (4y^5)^2 \]
Calculating each term:
- \((3x)^2 = 9x^2\)
- \(-2(3x)(4y^5) = -24xy^5\)
- \((4y^5)^2 = 16y^{10}\)
Putting it all together:
\[ (3x - 4y^5)^2 = 9x^2 - 24xy^5 + 16y^{10} \]
Therefore, we have:
\[ (3x - 4y^5)^{-2} = \frac{1}{9x^2 - 24xy^5 + 16y^{10}} \]
Now, let's look at the choices provided:
- \(-9x^8y^{10}\)
- \(\frac{x^8}{9y^{10}}\)
- \(\frac{1}{9x^8y^{10}}\)
The correct expression equivalent to \((3x - 4y^5)^{-2}\) simplifies to a fraction involving a squared expression, hence none of the options listed exactly match. However, if we were to consider a more suitable form based on combinations of x and y powers, we could explore additional simplification or factorization.
Unfortunately, from the choices given, none directly correspond with the result we simplified. Thus, we can conclude about the expressions, but it looks like there is an error with the provided answer choices as they don't relate directly to our derived expression. If you have additional context or information, please provide that for further exploration!
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