To simplify the given expression using the properties of integer exponents, let’s first clarify the notation for exponents.
It seems you have a complex expression with exponents, but let’s break it down step by step.
Given Expression:
(-5 a^(3) b^(1) c^(-2))^(3 a^(-4) b^(0))
Then, the expression outside the parentheses seems to be:
(-2)^(9/25) c^(12)/ (14) b^(-8)
Simplifying the expression in the parentheses:
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Exponent Rules:
- \( (x^m)^n = x^{m \cdot n} \)
- \( x^{-n} = \frac{1}{x^n} \)
- \( x^0 = 1 \)
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Applying the Exponential Rule: Let's simplify the expression inside the parentheses: \[ (-5 a^{3} b^{1} c^{-2})^{3a^{-4}b^{0}} \] Since \(b^{0} = 1\), we can focus on: \[ (-5 a^{3} c^{-2})^{3a^{-4}} \]
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Expanding Each Component:
- The constant: \[ (-5)^{3a^{-4}} = -5^{3a^{-4}} \]
- The \(a\) term: \[ a^{3 \cdot 3a^{-4}} = a^{9a^{-4}} = a^{9 / a^{4}} = a^{9 - 4} = a^{5} \]
- The \(c\) term: \[ c^{-2 \cdot 3a^{-4}} = c^{-6a^{-4}} = c^{(-6 / a^{4})} \]
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Putting It Together: The expression inside the parentheses becomes: \[ (-5)^{3a^{-4}} a^{5} c^{(-6 / a^{4})} \]
Now consider the expression outside:
If we assume the expression outside is to multiply or simplify against what we got inside, it becomes complex without more context.
In conclusion, without exact readability in the algebraic notation, it seems more complex than a straightforward fraction of integers with \(9c^{10}\) or \(-5c^{10}\).
Ultimately:
From the options provided (9C with a small 10 on top, -5C with a small tan on top, etc.), clarify the base structure or if you'd like to use numerical values for constants (-5, etc.) for better simplification or structure to give the correct resulting answer. Sometimes, due to the complexity of exponents and proper simplification, it helps to break each part down stepwise.
If you can provide clear details on how the expressions relate to each other or specific numerical evaluations, I'd be glad to assist further!
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