Question

In Exercises 69–82, simplify each complex rational expression. (1/(x+h)2 − 1/x2)/h

Accepted Answer

Step 1: Recognize that the given expression is a complex rational expression. A complex rational expression is a fraction where the numerator and/or denominator contains fractions. The given expression is $$(1/(x+h)^2$$ − $$1/x^2)/h. $$Step 2: Simplify the numerator of the complex fraction, which is $$1/(x+h)^2$$ − $$1/x^2. To do $$this, find a common denominator for the two fractions in the numerator. The least common denominator (LCD) of $$(x+h)^2$$ and $$x^2 is$$ $$(x+h)^2$$ * $$x^2. $$Step 3: Rewrite each fraction in the numerator with the LCD as the denominator. For $$1/(x+h)^2$$, multiply numerator and denominator by $$x^2 to $$get $$x^2/((x+h)^2$ * x^2$). For 1/x^2$, multiply numerator and denominator by (x+h)^2$ to get (x+h)^2$/((x+h)^2$$ * $$x^2). $$Step 4: Combine the two fractions in the numerator over the common denominator. This gives $$(x^2$$ - $$(x+h)^2$)/((x+h)^2$$ * $$x^2). $$Expand $$(x+h)^2 in $$the numerator to simplify further. Step 5: Divide the simplified numerator by h, which is the denominator of the original complex fraction. This is equivalent to multiplying the simplified numerator by 1/h. Simplify the resulting expression by factoring and canceling terms where possible.

In Exercises 69–82, simplify each complex rational expression. (1/(x+h)² − 1/x²)/h

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Key Concepts

Complex Rational Expressions

Finding a Common Denominator

Limits and Simplification