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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

1 answer

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

  1. 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:

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