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Chapter 0: Problem 76

Contain fractional expressions that occur frequently in calculus. Simplify each expression. $$\frac{\sqrt{5-x^{2}}+\frac{x^{2}}{\sqrt{5-x^{2}}}}{5-x^{2}}$$

Short Answer

Expert verified
The simplified version of the given expression is \( \sqrt{5-x^{2}}+1 \)

Step by step solution

01

Normalize the numerator

To normalize the numerator, multiply the first term by \(\sqrt{5-x^{2}}\) and the second term by \(5-x^{2}\) in the denominator. \[\frac{\sqrt{5-x^{2}}*(5-x^{2})+x^{2}}{5-x^{2}}\]
02

Simplify the numerator

Now simplify the numerator by multiplying the terms we got from Step 1: \[\frac{5\sqrt{5-x^{2}}-x^{2}\sqrt{5-x^{2}}+x^{2}}{(5-x^{2})}\]
03

Factorize the numerator

Factor the numerator by taking out common terms. Here, we can take \(\sqrt{5-x^2}\) common, which gives. \[\frac{\sqrt{5-x^{2}}(5-x^{2}+1)}{(5-x^{2})}\]
04

Simplify the expression

Now simplify the expression by canceling the common terms in the numerator and the denominator. After canceling, we get: \[\sqrt{5-x^{2}}+1\]
05

Finalize the Simplified Expression

The simplified version of the original expression is: \[\sqrt{5-x^{2}}+1\]

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Most popular questions from this chapter

How is the quadratic formula derived?

Perform the indicated operations. $$\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x+1}\right)\left(1-\frac{1}{x+2}\right)\left(1-\frac{1}{x+3}\right)$$

What is a compound inequality and how is it solved?

Will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

$$\text { Solve for } C: \quad V=C-\frac{C-S}{L} N$$.
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