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Chapter 0: Problem 79

Rationalize the numerator. $$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

Short Answer

Expert verified
The rationalized form of the given expression is \( \frac{1}{\sqrt{x+5} + \sqrt{x}} \).

Step by step solution

01

Identify the expression to rationalize

Identify the numerator of the fraction to be rationalized, which is \( \sqrt{x+5} - \sqrt{x} \). The goal is to eliminate the square roots from the numerator.
02

Utilize the difference of squares formula

To eliminate the square roots, we will use the principle of difference of squares formula, which states that for any two terms a and b, \( (a+b)(a-b) = a^2 - b^2 \). In our case the terms a and b are \( \sqrt{x+5} \) and \( \sqrt{x} \). So multiplying the given fraction by \( \frac{\sqrt{x+5} + \sqrt{x}}{\sqrt{x+5} + \sqrt{x}} \) will result in: \[\frac{(\sqrt{x+5}-\sqrt{x})(\sqrt{x+5}+\sqrt{x})}{5(\sqrt{x+5}+\sqrt{x})}\]
03

Simplify the fraction

Simplify the function by applying the difference of squares to the numerator and simplifying the denominator: \[ = \frac{((\sqrt{x+5})^2 - (\sqrt{x})^2)}{5(\sqrt{x+5}+\sqrt{x})}\] Further simplification leads to: \[= \frac{x+5 - x}{5(\sqrt{x+5}+\sqrt{x})} = \frac{5}{5(\sqrt{x+5}+\sqrt{x})} = \frac{1}{\sqrt{x+5} + \sqrt{x}}\]

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Most popular questions from this chapter

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathbf{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Explain how to determine the restrictions on the variable for the equation $$ \frac{3}{x+5}+\frac{4}{x-2}=\frac{7}{x^{2}+3 x-6} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Explain how to add or subtract rational expressions with the same denominators.
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