Question

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

−5c103a14b8
Start Fraction negative 5 c superscript 10 baseline over 3 a superscript 14 baseline b superscript 8 baseline End Fraction

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

Answer 1

To simplify the expression \((-5a^3b^4c^{-2} - 3a^{-4}b^0c^3)^{-2}\), we can follow these steps:

1. **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:
\[
-5a^3b^4c^{-2} - 3a^{-4}c^3
\]

2. **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.

3. **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.

4. **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.

5. **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})**.