Problem 79 - College Algebra

Question

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

Step-by-Step Solution

Answer

The simplified form is: \(\frac{a(c+d)}{{a^{3}-b^{3}}}-\frac{b(c+d)}{{a^{3}-b^{3}}}-\frac{c-d}{{a^{2}+a b+b^{2}}}\)

Step 1: Simplify the Equation

First, we look for any immediate simplifications that we can apply. Given that the denominator 1 in the expression \(\frac{{a c+a d-b c-b d}}{1}\) does not affect the value of the expression, we can rewrite it as \(a c+a d-b c-b d\).

Step 2: Apply Distributive Property

We notice that we can factor out an \(a\) from \(a c+a d\) and a \(b\) from \(-b c-b d\). We then apply the distributive property and thus get: \(\frac{1}{{a^{3}-b^{3}}}\cdot (a(c+d)-b(c+d))\).

Step 3: Distribute the Reciprocal

Next, to simplify the expression further, we distribute the reciprocal throughout the two parts of the expression in parentheses. The resulting equation becomes: \(\frac{a(c+d)}{{a^{3}-b^{3}}}-\frac{b(c+d)}{{a^{3}-b^{3}}}\).

Step 4: Evaluate the Minus Expression

Next, we simplify the expression \(\frac{c-d}{{a^{2}+a b+b^{2}}}\). It remained unchanged as there’s no simplification beyond this.

Step 5: Final Expression

Combining the results from Step 3 and Step 4, we get the final, simplified expression: \(\frac{a(c+d)}{{a^{3}-b^{3}}}-\frac{b(c+d)}{{a^{3}-b^{3}}}-\frac{c-d}{{a^{2}+a b+b^{2}}}\).