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Chapter 0: Problem 89

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

Short Answer

Expert verified
The simplified result is \(a - b - c + d\)

Step by step solution

01

Expand the expression

Considering the first term of the given expression, \(\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)\), we can simplify the multiplication by factorizing out the common factors, likewise in the subtraction operation \(\frac{c-d}{a^{2}+a b+b^{2}}\), we can rewrite it as: \(\frac{a*(c+d) -b*(c+d)}{a^{3}-b^{3}} - \frac{c-d}{a^{2}+a b+b^{2}}\)
02

Factorize

Notice that in the first fraction, there is a common factor of \((c+d)\) in the numerator and \((a^3 - b^3)\) in the denominator. Thus the product can be factored as: \(\frac{(a-b)(c+d)}{a^{3}-b^{3}} - \frac{c-d}{a^{2}+a b+b^{2}}\)
03

Simplify

Observe that in the first expression the denominator can be rewritten as \((a-b)*(a^2+ab+b^2)\), so the first term can be simplified to: \((a-b)\), similarly the second term, its denominator, is the same as in the first term and the first term numerator can be rewritten, hence this term simplifies to \((c-d)\). Hence when we subtract these terms we arrive at \((a-b)-(c-d)\)
04

Resolve the expression

Resolve \((a-b)-(c-d)\) to a simpler form. The result is \(a - b - c + d\)

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