Question

Write the rational expression in lowest terms.
StartFraction ak minus av plus 4 k minus 4 v Over ak minus av minus 4 k plus 4 v EndFraction

Answer 1

To simplify the rational expression

\[
\frac{ak - av + 4k - 4v}{ak - av - 4k + 4v}
\]

we first reorganize and combine like terms in both the numerator and the denominator.

### Step 1: Simplifying the Numerator

1. The numerator is:
\[
ak - av + 4k - 4v
\]
We can rearrange this as:
\[
(ak + 4k) + (-av - 4v) = k(a + 4) - v(a + 4)
\]
Thus, we can factor out \( (a + 4) \):
\[
= (k - v)(a + 4)
\]

### Step 2: Simplifying the Denominator

2. The denominator is:
\[
ak - av - 4k + 4v
\]
We can rearrange this as:
\[
(ak - 4k) + (-av + 4v) = k(a - 4) + v(-a + 4) = k(a - 4) + v(4 - a)
\]
Factoring gives:
\[
= (k + v)(a - 4)
\]

### Final Expression

Now that we have factored both the numerator and the denominator, we can rewrite the entire expression as:

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

### Step 3: Check for Common Factors

There do not seem to be any common factors between the numerator and the denominator. Therefore, we cannot simplify further.

### Result

The rational expression in lowest terms is:

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