To simplify the expression \((−5a^3b^4c−23a^{-4}b^0c^3)^{-2}\), we will follow the properties of integer exponents step by step.
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Simplify inside the parentheses: \(-5a^3b^4c - 23a^{-4}b^0c^3\)
- Since \(b^0 = 1\), the second term simplifies to \(-23a^{-4}c^3\).
- Thus, we have:
\(-5a^3b^4c - 23a^{-4}c^3\).
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Combine the terms:
The expression cannot be simplified further as-is since they have different variable powers. So we keep it as it is. -
Raise the whole expression to the power of -2:
\((−5a^3b^4c - 23a^{-4}c^3)^{-2}\). -
Distributing the negative exponent:
When raising an expression to a negative power, we take the reciprocal of the expression and change the sign of the exponent: \[\frac{1}{(-5a^3b^4c - 23a^{-4}c^3)^{2}}\] -
Consider the expression in the denominator:
To further analyze:- Simplify terms if applicable by factoring out common bases if possible.
- However, as given, this expression remains complex.
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Analyzing the terms:
But for simplifying specific terms after expanding, we will expand \((-5a^3b^4c)\) and \(-23a^{-4}c^3\) if writings appear quadratic: We can calculate:- \((-5a^3b^4c)^2\) gives \(25a^6b^8c^2\).
- \((-23a^{-4}c^3)^2\) gives \(529a^{-8}c^6\).
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Adding or handling the distribution of squares gives: Hence \(= 25a^6b^8c^2 + 529a^{-8}c^6\).
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Final result:
Therefore, the whole expression raised will give: \[\frac{1}{25a^6b^8c^2 + 529a^{-8}c^6}\] retain the basic structure on final returns.
However, among the response choices given, the closest to any manipulation returning the values in certain forms would rotate back down as common-base separation designed.
Thus the correct answer among your options would be:
c10a14b8 \[ \frac{c^{10}}{a^{14}b^{8}} \]
Please confirm if further simplifications can clear or should analyze another route of base handling if needed!