/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Contain fractional expressions t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 77

Contain fractional expressions that occur frequently in calculus. Simplify each expression. $$\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}$$

Short Answer

Expert verified
\(\frac{2}{\sqrt{x(x+h)}}\)

Step by step solution

01

Rationalizing the numerator

To make the expression easier to handle, first rationalize the numerator of the given complex fraction. This can be done by multiplying both the numerator and the denominator by the conjugate of the numerator. The conjugate of \( \frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}} \) is \( \frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}} \): \[\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h} * \frac{\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}}\]
02

Multiplying through

Multiplication of the fractions yields: \[\frac{ \left(\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}\right) * \left(\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}\right) }{ h * \left(\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}\right) }\]
03

Simplification

Perform the indicated operation in the numerator and denominator. To do this, apply the formula \(a^2-b^2=(a+b)(a-b)\) in the numerator. In the denominator, distribute the term h. Then we get: \[\frac{ \frac{(x+2h)+x}{x(x+h)} - \frac{1}{x} }{ h * \left(\frac{1}{\sqrt{x+h}}+\frac{1}{\sqrt{x}}\right) }\] Simplify to get: \[\frac{2}{\sqrt{x+h} + \sqrt{x}}\]
04

Simplification

Simplify further: \[\frac{2}{\sqrt{x+h} + \sqrt{x}}=\frac{2\sqrt{x} \sqrt{x+h}}{x(x+h)}=\frac{2x\sqrt{x+h}}{x(x+h)} \] Consequently, the simplified expression is: \[ \frac{2}{\sqrt{x(x+h)}} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

Describe ways in which solving a linear inequality is similar to solving a linear equation.

This will help you prepare for the material covered in the next section. Evaluate $$\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$ for \(a=2, b=9,\) and \(c=-5\)

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.