Apply the properties of integer exponents to simplify the algebraic expression (−5a3b4c−23a−4b0c3)−2 .(1 point)

Question

Responses

9c1025a14b8
Start Fraction 9 c superscript 10 baseline over 25 a superscript 14 baseline b superscript 8 baseline End Fraction

925a12
Start Fraction 9 over 25 a superscript 12 baseline End Fraction

c10a14b8
Start Fraction c superscript 10 baseline over a superscript 14 baseline b superscript 8 baseline End Fraction

−5c103a14b8

Answer 1

To simplify the expression \((−5a^3b^4c−23a^{-4}b^0c^3)^{-2}\), we first simplify the expression inside the parentheses before applying the exponent.

1. Simplify the expression inside the parentheses:

   • \( b^0 = 1 \) (any variable raised to the power of 0 is 1),
   • Therefore, the expression inside becomes: \[ -5a^3b^4c - 23a^{-4} \cdot 1 \cdot c^3 = -5a^3b^4c - 23a^{-4}c^3. \]

2. Factor out the common terms:

   • The expressions \(-5a^3b^4c\) and \(-23a^{-4}c^3\) do not have common factors other than \(c\), which we can express as: \[ c(-5a^3b^4 - 23a^{-4}c^2). \]

3. Raising to the power of -2: \[ \left(c(-5a^3b^4 - 23a^{-4}c^2)\right)^{-2} = c^{-2}(-5a^3b^4 - 23a^{-4}c^2)^{-2}. \]

4. Performing the exponent on both parts:

   • We know \(c^{-2} = \frac{1}{c^2}\).
   • Now we need to handle the expression \((-5a^3b^4 - 23a^{-4}c^2)^{-2}\). Since this expression is not a straightforward simplification at this stage, we can treat it as a whole.

5. Expanding and simplifying the full expression:

   • The expression becomes \(\frac{1}{c^2} \cdot \frac{1}{(-5a^3b^4 - 23a^{-4}c^2)^2}\).
   • This is a complex polynomial expression, but we will proceed generally as \(c\) does not define it better without context.

Thus, we're looking to simplify the output expression generally, leading us to examine the output options:

Without doing an exhaustive expansion of the polynomial (which might be too complex without specific guidance), the most reasonable output “fits” would be to simplify to an expression that shows the general combinations of \(a\), \(b\), and \(c\) based on equivalent conditions shown. Since we don't have enough to factor singly, let's review the choices presented:

Given Responses Evaluating Choices with Final Choices:

• The direct simplification appears rather abstract without evaluating contradictions.
• Nevertheless, focusing on output types from overall reasoning, we find only "cleaned" options.

The most suitable form that seems reasonable from the provided choices must equate to expressions with \(c\) primarily denoting powers from the structure indicated \( \frac{c^{10}}{a^{14}b^8}.\)

Concisely looking through all factors listed, this leads to the final clean process where the rate of combinations will give:

Final Choice: \( \frac{c^{10}}{a^{14}b^8} \).