To simplify the expression \((-5a^3b^4c^{-2} - 3a^{-4}b^0c^3)^{-2}\), we can follow these steps:
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Simplify each term inside the parentheses:
- The first term is \(-5a^3b^4c^{-2}\).
- The second term is \(-3a^{-4}b^0c^3\). Since \(b^0 = 1\), it simplifies to \(-3a^{-4}c^3\).
So, we have: \[ -5a^3b^4c^{-2} - 3a^{-4}c^3 \]
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Combine the terms by finding a common denominator, if necessary: To combine these two terms, we need them in the same form. The common denominator for the \(c\) terms would be \(c^{-2}\) and \(c^3\), which translates to: \[ -5a^3b^4 + 3a^{-4}c^5 \] This is already quite complex, but we will raise the entire expression to the power of \(-2\) next.
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Factor and simplify: We rewrite the expression: \[ = (-5a^3b^4c^{-2} - 3a^{-4}c^{3})^{-2} \]
Raise each term inside parentheses separately to the power of \(-2\), and then apply the properties of exponents.
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Apply the exponent: The expression becomes: \[ \frac{1}{(-5a^3b^4c^{-2} + 3a^{-4}c^{3})^{2}} \]
In essence, we need the negative exponent law which tells that \(x^{-n} = \frac{1}{x^n}\) and simplifying gives details of how \(c^{-2}\) and powers of \(a\) and \(b\) interact.
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Final Cleaning and Simplifying: Compute all coefficients: \[ = \frac{1}{25}c^{10}a^{14}b^8 \text{ (to the respective powers)}. \]
So, ultimately, our expression should yield: \[ \frac{9c^{10}}{25a^{14}b^{8}} \]
Thus, the correct response is: 9c^{10} / (25a^{14}b^{8}).