https://file.aiquickdraw.com/mathbot/file/1725455901385c002k26i.jpg

The expression provided in the image is:

\[
\frac{a^{-2} - 4}{b^{-2} - b^{-1} - 2}
\]

Let's simplify this step by step.

### Step 1: Simplify the numerator \( a^{-2} - 4 \)
- The term \( a^{-2} \) is equivalent to \( \frac{1}{a^2} \).
- So, the numerator becomes:
  \[
  \frac{1}{a^2} - 4
  \]

### Step 2: Simplify the denominator \( b^{-2} - b^{-1} - 2 \)
- \( b^{-2} \) is equivalent to \( \frac{1}{b^2} \).
- \( b^{-1} \) is equivalent to \( \frac{1}{b} \).
- So, the denominator becomes:
  \[
  \frac{1}{b^2} - \frac{1}{b} - 2
  \]

### Step 3: Write the simplified expression
Now, the expression can be written as:
\[
\frac{\frac{1}{a^2} - 4}{\frac{1}{b^2} - \frac{1}{b} - 2}
\]

### Step 4: Find a common denominator and simplify further
Let's factor the expression further if possible. For simplicity, let's observe if we can factorize the numerator and denominator to find any common terms. However, in this specific form, the expression is in its simplified state without specific values for \(a\) and \(b\). 

**Result**:
\[
\frac{\frac{1}{a^2} - 4}{\frac{1}{b^2} - \frac{1}{b} - 2}
\]

This is the simplified expression.

Would you like more details, or do you have any questions? Here are some related questions:

1. How would you simplify an expression involving negative exponents further if given specific values?
2. Can you factor the denominator more effectively in this expression?
3. What happens if \( a = 1 \) and \( b = 1 \) in the simplified expression?
4. How would the expression change if \( b^{-2} - b^{-1} \) were the only terms in the denominator?
5. What are the steps to evaluate this expression for given values of \( a \) and \( b \)?

**Tip:** When simplifying expressions involving exponents, always remember to handle negative exponents by converting them into fractions.

Simplify the expression: (a^(-2) - 4) / (b^(-2) - b^(-1) - 2)

Learn how to simplify expressions involving negative exponents and rational expressions. This problem breaks down the steps to simplify (a^(-2) - 4) / (b^(-2) - b^(-1) - 2) by handling negative exponents and combining terms. Suitable for students in Grades 9-11.

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