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Chapter 0: Problem 75

Contain fractional expressions that occur frequently in calculus. Simplify each expression. $$\frac{\frac{x^{2}}{\sqrt{x^{2}+2}}-\sqrt{x^{2}+2}}{x^{2}}$$

Short Answer

Expert verified
The simplified form of given expression is \(\frac{1}{\sqrt{x^{2}+2}} - \frac{\sqrt{x^{2}+2}}{x^2}\)

Step by step solution

01

Simplification

Initially, take the given expression: \(\frac{\frac{x^{2}}{\sqrt{x^{2}+2}}-\sqrt{x^{2}+2}}{x^{2}}\). Observe that it is in the form of \(\frac{a - b}{c}\) which can be split into \(\frac{a}{c} - \frac{b}{c}\) . Let's apply that in order to simplify this expression.
02

Split the expression

Split the expression into two fractions as indicated above: \(\frac{\frac{x^{2}}{\sqrt{x^{2}+2}}}{x^{2}} - \frac{\sqrt{x^{2}+2}}{x^{2}}\). Now apply the division rule, that is \(\frac{a}{b} * \frac{c}{d} = \frac{a*c}{b*d}\).
03

Simplification

Apply the division rule from the previous step for the first part of the split expression, \(\frac{x^{2}}{\sqrt{x^{2}+2}} * \frac{1}{x^{2}} = \frac{x^{2}}{x^{2}*\sqrt{x^{2}+2}}\). After simplifying this you get, \(\frac{1}{\sqrt{x^{2}+2}}\). Apply the rule \(\frac{1}{\sqrt{a^2}} = \frac{1}{a}\), to get: \(\frac{1}{\sqrt{x^{2}+2}}\) which is the first part of the answer.
04

Simplification

Let's now solve the second part of the split expression: \(\frac{\sqrt{x^{2}+2}}{x^{2}}\). Apply the rule for expressing in terms of square root, \(\frac{\sqrt{a}}{b^2} = \frac{\sqrt{a}}{b*\sqrt{b}}\), which gives: \(\frac{\sqrt{x^{2}+2}}{x*\sqrt{x}}\), which simplifies to \(\frac{\sqrt{x^{2}+2}}{x^2}\).
05

Combining the Parts

Combine the calculated expressions from step 3 and step 4. So, the final answer becomes: \(\frac{1}{\sqrt{x^{2}+2}} - \frac{\sqrt{x^{2}+2}}{x^2}\)

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3,00\) to produce cach package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x^{2}+36=(x+6)^{2}$$

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