/*! 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 72 Rationalize each numerator. If p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 72

Rationalize each numerator. If possible, simplify your result. $$\frac{\sqrt{3}+1}{4}$$

Short Answer

Expert verified
\[ \frac{1}{2(\sqrt{3} - 1)} \]

Step by step solution

01

- Identify the Conjugate

To rationalize the numerator, first identify the conjugate of the numerator. The numerator is \( \sqrt{3} + 1 \). The conjugate of \( \sqrt{3} + 1 \) is \( \sqrt{3} - 1 \).
02

- Multiply Numerator and Denominator

Multiply both the numerator and the denominator by the conjugate \( \sqrt{3} - 1 \): \[\frac{( \sqrt{3} + 1)( \sqrt{3} - 1 )}{4(\sqrt{3} - 1)}\]
03

- Apply Difference of Squares

Simplify the numerator using the difference of squares formula, \(a^2 - b^2\): \[(\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2\]
04

- Simplify the Denominator

Now, simplify the fraction: \[\frac{2}{4(\sqrt{3} - 1)} = \frac{2}{4(\sqrt{3} - 1)} = \frac{1}{2(\sqrt{3} - 1)}\]
05

- Simplify the Expression

Final form of the simplified expression is: \[\frac{1}{2(\sqrt{3} - 1)}. \] This is the simplest form of the given expression.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

conjugate
When rationalizing the numerator, the first step is to identify the conjugate. The conjugate of a binomial expression changes the sign in the middle. For example, the conjugate of \( \sqrt{3} + 1 \) is \( \sqrt{3} - 1 \). Using the conjugate helps to eliminate the square root in the numerator. This step is crucial because it transforms the original problem in a way that makes further simplification possible.

Conjugates are a pair of expressions that, when multiplied, produce a difference of squares. This is useful because the square root terms cancel out, leaving a simpler result. Identifying the conjugate correctly is the first important milestone in this process.
difference of squares
The difference of squares is a formula used to simplify expressions. This formula states: \[ a^2 - b^2 = (a + b)(a - b) \]. Applying this to our problem, we use the conjugates we identified earlier. Multiplying \( \sqrt{3} + 1 \) by \( \sqrt{3} - 1 \), we get:

\[ (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2 \].

The difference of squares helps us to get rid of the square roots in the numerator, simplifying it to just 2. This transformation is a pivotal step as it paves the way for further simplification of the expression.
simplification
Simplification makes expressions easier to understand and work with. After applying the difference of squares, the expression becomes much simpler, and we can focus on the denominator. With the numerator simplified to 2, we now have:

\[ \frac{2}{4(\sqrt{3} - 1)} \].

Notice that both 2 and 4 in the fraction can be further simplified. So the expression can be written as: \[ \frac{1}{2(\sqrt{3} - 1)} \].

This final step consolidates all previous efforts into a simplified, rationalized form. Simplifying might also include canceling out common factors, combining like terms, or reducing fractions to their simplest forms, which makes solving and understanding the problem easier.

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

Simplify. $$\frac{\frac{1}{\sqrt{w}}-\sqrt{w}}{\frac{\sqrt{w}+1}{\sqrt{w}}}$$

To prepare for Section \(10.6,\) review solving equations (Sections 2.2 and 7.6 and Chapter 6 ). Solve. $$9 x^{2}-6 x+1=7+5 x-x^{2}$$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{2} \sqrt[3]{4} $$

Multiply. $$(\sqrt{x+2}-\sqrt{x-2})^{2}$$

In what way(s) is combining like radical terms similar to combining like terms that are monomials?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics 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.