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Chapter 0: Problem 72

Simplify each complex rational expression. $$\frac{\frac{x+h}{x+h+1}-\frac{x}{x+1}}{h}$$

Short Answer

Expert verified
The complex rational expression simplifies to: \( \frac{1}{(x+1)(x+h+1)}\)

Step by step solution

01

Applying common denominators

The first thing to do in simplifying this complex fraction is to make the denominators the same for the fractions \(\frac{x+h}{x+h+1}\) and \(\frac{x}{x+1}\). We multiply the first fraction by \(\frac{x+1}{x+1}\) and the second fraction by \(\frac{x+h+1}{x+h+1}\). This will give us: \(\frac{\frac{(x+h)(x+1)}{(x+h+1)(x+1)}-\frac{x(x+h+1)}{(x+1)(x+h+1)}}{h}\)
02

Simplifying the numerator

Now eliminate the fractions in the numerator by applying the difference of fractions (subtract the numerators and keep the common denominator). It becomes: \(\frac{[(x+h)(x+1)] - [x(x+h+1)]}{(x+1)(x+h+1)h}\)
03

Adding, multiplying and reducing

After applying difference of fractions and multiplication: \(\frac{x^2+xh+x+h-x^2-xh-x}{(x+1)(x+h+1)h}\) This simplifies down to: \( \frac{h}{(x+1)(x+h+1)h}\)
04

Further Simplify

In the current fraction, you see that there's a 'h' in the numerator and the denominator. We can simplify this by dividing top and bottom by 'h', yielding: \( \frac{1}{(x+1)(x+h+1)}\)

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Most popular questions from this chapter

Explain how to determine which numbers must be excluded from the domain of a rational expression.

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