Perform the indicated operations.

Question

StartFraction a plus b Over 72 a squared plus 23 ab minus 4 b squared EndFraction plus StartFraction 4 a minus b Over 64 a squared minus b squared EndFraction

Answer 1

To add the two fractions:

\[ \frac{a + b}{72a^2 + 23ab - 4b^2} + \frac{4a - b}{64a^2 - b^2} \]

we need both fractions to have a common denominator. First, let's factor the denominators.

Factor the Denominators

1. Denominator 1: \(72a^2 + 23ab - 4b^2\) We can use the factoring method for trinomials or the quadratic formula to find factors. However, let's assume we check for factors and find:

   \[ 72a^2 + 23ab - 4b^2 = (8a - b)(9a + 4b) \]

2. Denominator 2: \(64a^2 - b^2\) This is a difference of squares: \[ 64a^2 - b^2 = (8a - b)(8a + b) \]

Common Denominator

The common denominator for the two fractions will be the product of the distinct factors:

\[ (8a - b)(9a + 4b)(8a + b) \]

Rewrite Each Fraction

Now, let’s rewrite each fraction with the common denominator:

1. First Fraction \[ \frac{a + b}{(8a - b)(9a + 4b)} \cdot \frac{(8a + b)}{(8a + b)} = \frac{(a + b)(8a + b)}{(8a - b)(9a + 4b)(8a + b)} \]

2. Second Fraction \[ \frac{4a - b}{(8a - b)(8a + b)} \cdot \frac{(9a + 4b)}{(9a + 4b)} = \frac{(4a - b)(9a + 4b)}{(8a - b)(9a + 4b)(8a + b)} \]

Combine the Fractions

Now we can add the fractions:

\[ \frac{(a + b)(8a + b) + (4a - b)(9a + 4b)}{(8a - b)(9a + 4b)(8a + b)} \]

Expand the Numerator

1. First Term: \[ (a + b)(8a + b) = 8a^2 + ab + 8ab + b^2 = 8a^2 + 9ab + b^2 \]

2. Second Term: \[ (4a - b)(9a + 4b) = 36a^2 + 16ab - 9ab - 4b^2 = 36a^2 + 7ab - 4b^2 \]

Now we combine them:

\[ 8a^2 + 9ab + b^2 + 36a^2 + 7ab - 4b^2 = (8a^2 + 36a^2) + (9ab + 7ab) + (b^2 - 4b^2) \] \[ = 44a^2 + 16ab - 3b^2 \]

Final Answer

So the combined fraction is:

\[ \frac{44a^2 + 16ab - 3b^2}{(8a - b)(9a + 4b)(8a + b)} \]

This is the result of the indicated operations.