Question

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

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 of \(-2\).

First, recall that \(b^0 = 1\), so we can rewrite \(b^0\) in our expression:
\[
(-5a^3b^4c - 23a^{-4}b^0c^3) = (-5a^3b^4c - 23a^{-4}c^3)
\]

Now, we have:
\[
= -5a^3b^4c - 23a^{-4}c^3
\]

### Step 1: Combine like terms and factor if possible
There doesn't appear to be a straightforward way to combine these terms or factor them out because they have different powers of \(a\) and \(b\). Hence, we will directly consider each term as we apply the exponent.

### Step 2: Raise the entire expression to the power of \(-2\)
Next, we apply the exponent \(-2\) to the whole expression:
\[
(-5a^3b^4c - 23a^{-4}c^3)^{-2}
\]

### Step 3: Apply the exponent
Using the property \((x + y)^{-n} = \frac{1}{(x + y)^n}\), we can write:
\[
= \frac{1}{(-5a^3b^4c - 23a^{-4}c^3)^{2}}
\]

### Step 4: Expand the squared term
Now we need to compute the square of \( -5a^3b^4c - 23a^{-4}c^3 \):
\[
(-5a^3b^4c - 23a^{-4}c^3)^2 = (-5a^3b^4c)^2 + 2(-5a^3b^4c)(-23a^{-4}c^3) + (-23a^{-4}c^3)^2
\]

Calculating each piece:
1. \((-5a^3b^4c)^2 = 25a^6b^8c^2\)
2. \(2(-5a^3b^4c)(-23a^{-4}c^3) = 230a^{-1}b^4c^4\)
3. \((-23a^{-4}c^3)^2 = 529a^{-8}c^6\)

Combining these gives:
\[
(-5a^3b^4c - 23a^{-4}c^3)^2 = 25a^6b^8c^2 + 230a^{-1}b^4c^4 + 529a^{-8}c^6
\]

### Step 5: Put everything together
Now substitute back into our expression:
\[
(-5a^3b^4c - 23a^{-4}c^3)^{-2} = \frac{1}{25a^6b^8c^2 + 230a^{-1}b^4c^4 + 529a^{-8}c^6}
\]

Thus, the simplified expression is:
\[
\frac{1}{25a^6b^8c^2 + 230a^{-1}b^4c^4 + 529a^{-8}c^6}
\]