/*! 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 47 Simplify the expression. $$\fr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 47

Simplify the expression. $$\frac{12}{7-\sqrt{3}}$$

Short Answer

Expert verified
The simplified expression is \( \frac{42 + 6\sqrt{3}}{23} \).

Step by step solution

01

Identify the Conjugate

In this step, identify the conjugate of the denominator of the given expression. The conjugate of \(7-\sqrt{3}\) is \(7 + \sqrt{3}\).
02

Multiply by the Conjugate

In this step, multiply the numerator and the denominator by the conjugate. So, \(\frac{12}{7 - \sqrt{3}}\) multiply by \(\frac{7 + \sqrt{3}}{7 + \sqrt{3}}\) to get \(\frac{12 (7 + \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}\).
03

Simplify the Expression

In this step, simplify the expression by applying the difference of squares formula in the denominator which states that \(a^2 - b^2 = (a - b)(a + b)\), and distribute the 12 in the numerator. This gives \(\frac{84 + 12\sqrt{3}}{49 - 3} = \frac{84 + 12\sqrt{3}}{46}\).
04

Further Simplify

The numerator and the denominator have a common factor 2, so we can divide both sides by 2. This gives the simplified expression \(\frac{42 + 6\sqrt{3}}{23}\).

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.

Conjugates
When working with expressions that include roots in the denominator, one method to simplify the expression is to "rationalize the denominator." This often involves using the concept of conjugates. A conjugate is formed by changing the sign between two terms. For example, the conjugate of the expression
  • \( 7 - \sqrt{3} \) is \( 7 + \sqrt{3} \).

Multiplying the original expression's numerator and denominator by this conjugate helps eliminate the square root from the denominator. This is because when a binomial expression, such as \((a - b)\), is multiplied by its conjugate, \((a + b)\), the result is a difference of squares.
Simplifying Expressions
Simplifying mathematical expressions is a crucial step in solving equations and making calculations easier. In our example, we simplify \( \frac{12(7+\sqrt{3})}{(7-\sqrt{3})(7+\sqrt{3})} \) by distributing the numerator and applying algebraic identities.
  • First, distribute the 12 across the terms inside the parenthesis: \( 12 \times 7 + 12 \times \sqrt{3} \).
  • This results in \( 84 + 12\sqrt{3} \).
It's essential to simplify expressions because it makes them easier to understand and work with, especially when solving equations. In many cases, after distribution and simplification, you might notice common factors that allow further reduction of the expression.
Difference of Squares
The difference of squares is a specific algebraic pattern used to simplify expressions like the ones we encounter when rationalizing denominators. The identity is given by:
  • \( a^2 - b^2 = (a - b)(a + b) \)

Applying this rule to
  • \( (7 - \sqrt{3})(7 + \sqrt{3}) \)
results in a simpler expression:
  • \( 7^2 - (\sqrt{3})^2 = 49 - 3 = 46 \).
Understanding this pattern is essential when dealing with polynomial expressions and helps in simplifying rational expressions by eliminating radicals, streamlining calculations, and leading the way to a more straightforward solution.

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

Decide whether the ordered pair is a solution of the inequality. $$y \leq 2 x^{2}-3 x+10 ;(-2,20)$$

Find the midpoint between the two points \((5,-5),(-5,1)\)

Two numbers and their geometric mean are given. Find the value of \(a\). 4 and \(a ; 16\)

Multiply. $$(2 a-9 b)^{2}$$

Use a graphing calculator to graphically solve the radical equation. Check the solution algebraically. $$\sqrt{15-4 x}=2 x$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.