https://file.aiquickdraw.com/mathbot/file/1727056769899oiu9a2f8.jpg

Let's simplify the expression given in the image step by step. The expression appears to be:

\[
\frac{\frac{16a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{4}{b}}
\]

### Step 1: Simplify the Numerator
The numerator is \(\frac{16a}{b} - \frac{b}{a}\).

To subtract these fractions, we need a common denominator. The common denominator between \(b\) and \(a\) is \(ab\). So we rewrite each fraction:

\[
\frac{16a}{b} = \frac{16a^2}{ab}, \quad \frac{b}{a} = \frac{b^2}{ab}
\]

Now, subtract them:

\[
\frac{16a^2}{ab} - \frac{b^2}{ab} = \frac{16a^2 - b^2}{ab}
\]

### Step 2: Simplify the Denominator
The denominator is \(\frac{1}{a} - \frac{4}{b}\).

Again, we need a common denominator, which is \(ab\). Rewriting each fraction:

\[
\frac{1}{a} = \frac{b}{ab}, \quad \frac{4}{b} = \frac{4a}{ab}
\]

Now, subtract them:

\[
\frac{b}{ab} - \frac{4a}{ab} = \frac{b - 4a}{ab}
\]

### Step 3: Form the Overall Fraction
Now we have the expression:

\[
\frac{\frac{16a^2 - b^2}{ab}}{\frac{b - 4a}{ab}}
\]

When dividing fractions, we multiply by the reciprocal of the denominator. So we get:

\[
\frac{16a^2 - b^2}{ab} \times \frac{ab}{b - 4a}
\]

The \(ab\) terms cancel out, leaving us with:

\[
\frac{16a^2 - b^2}{b - 4a}
\]

### Step 4: Factor the Numerator
The numerator \(16a^2 - b^2\) is a difference of squares, which can be factored as:

\[
16a^2 - b^2 = (4a - b)(4a + b)
\]

### Final Expression
Now we have:

\[
\frac{(4a - b)(4a + b)}{b - 4a}
\]

Notice that \(b - 4a\) is the same as \(-(4a - b)\). So, we can cancel out the \((4a - b)\) terms, but we are left with a negative sign:

\[
-(4a + b)
\]

Thus, the simplified expression is:

\[
-(4a + b)
\]

### Final Answer:
\[
-(4a + b)
\]

Would you like further details or have any questions?

---

Here are five questions for further exploration:
1. What is the importance of finding a common denominator in fraction operations?
2. How can you identify the difference of squares when simplifying expressions?
3. Why does canceling terms in a rational expression require factoring first?
4. What are the conditions for canceling terms in algebraic fractions?
5. How does the negative sign affect the result when simplifying fractions?

**Tip:** Always check if terms in the numerator or denominator can be factored, as it often simplifies the process of solving.

Simplify the expression: (16a/b - b/a) / (1/a - 4/b).

Learn how to simplify the rational expression (16a/b - b/a) / (1/a - 4/b). This problem involves finding a common denominator and factoring the difference of squares. Step-by-step solution provided, suitable for students in Grades 9-12.

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