/*! 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 55 Simplify. $$\frac{\sqrt{3}}{\s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 55

Simplify. $$\frac{\sqrt{3}}{\sqrt{x}+8}$$

Short Answer

Expert verified
\( \frac{\sqrt{3} \sqrt{x} - 8\sqrt{3}}{x - 64} \)

Step by step solution

01

Rationalizing the denominator

To simplify the expression \(\frac{\sqrt{3}}{\sqrt{x} + 8}\), multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x} - 8\). This helps to eliminate the square root in the denominator. The expression becomes: \[\frac{\sqrt{3}}{\sqrt{x} + 8} \cdot \frac{\sqrt{x} - 8}{\sqrt{x} - 8} = \frac{\sqrt{3} (\sqrt{x} - 8)}{(\sqrt{x} + 8)(\sqrt{x} - 8)}\]
02

Simplify the denominator

Use the difference of squares formula \((a + b)(a - b) = a^2 - b^2\) to simplify the denominator: \[ (\sqrt{x} + 8)(\sqrt{x} - 8) = (\sqrt{x})^2 - 8^2 = x - 64 \] So the expression now is: \[ \frac{\sqrt{3} (\sqrt{x} - 8)}{x - 64} \]
03

Distribute in the numerator

Distribute \(\sqrt{3}\) in the numerator: \[ \sqrt{3} (\sqrt{x} - 8) = \sqrt{3} \sqrt{x} - 8\sqrt{3} \] Thus, the expression simplifies to: \[ \frac{\sqrt{3} \sqrt{x} - 8\sqrt{3}}{x - 64} \]

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.

Rationalizing the Denominator
When faced with a fraction that has a square root in the denominator, simplifying the expression often involves a process called rationalizing the denominator.
The goal is to remove the square root so that the denominator is a rational number.
To achieve this, we use the conjugate of the denominator. The conjugate means changing the sign between two terms in a binomial.

Let's break down our original exercise: \(\frac{\sqrt{3}}{\sqrt{x} + 8}\).
We multiply both the numerator and the denominator by the conjugate of the denominator: \(\sqrt{x} - 8\).
This leads to:
  • Numerator: \(\frac{\sqrt{3} (\sqrt{x} - 8)}{(\sqrt{x} + 8)(\sqrt{x} - 8)}\)
  • Denominator: The product of the denominator and its conjugate.
Using the conjugate allows us to clear the square root, making the expression much simpler to work with.
Difference of Squares
In algebra, the difference of squares is a useful pattern to recognize.
It simplifies expressions where you have the form \(a^2 - b^2\).
This form always factors into \(a + b\)(\a - b\).

For our original problem, when we multiplied \(\sqrt{x} + 8\) and its conjugate \(\sqrt{x} - 8\), we use the difference of squares formula:
  • \( (\sqrt{x} + 8)(\sqrt{x} - 8) = (\sqrt{x})^2 - 8^2 = x - 64)
Using the difference of squares helps transform the denominator to get rid of the square root, giving us:
\(\frac{\sqrt{3} (\sqrt{x} - 8)}{x - 64}\), which is more straightforward to handle.
Distribution in Algebra
Distribution in algebra involves multiplying each term inside a parenthesis by a term outside the parenthesis.
In our scenario, we need to distribute the \(\sqrt{3}\) in the numerator.
This is called the distributive property and ensures all terms are appropriately multiplied:
  • Original Numerator: \(\sqrt{3} (\sqrt{x} - 8)\)
  • Distribute \(\sqrt{3}\): \(\sqrt{3} \cdot \sqrt{x} - \sqrt{3} \cdot 8\)
When we distribute, the expression simplifies to:
\(\sqrt{3} \sqrt{x} - 8\sqrt{3} \).
This gives us the final simplified version of the expression:
\(\frac{\sqrt{3} \sqrt{x} - 8\sqrt{3}}{x - 64} \).

By distributing correctly, we ensure the algebraic expression is fully simplified and easy to understand.

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. $$8 \sqrt[4]{z}-3 \sqrt[4]{z}$$

The length of a rectangle is \(3 \mathrm{ft}\) more than twice its width. The perimeter of the rectangle is \(66 \mathrm{ft}\). Use a system of linear equations to find its length and width.

Simplify. $$\sqrt[3]{40}$$

Evaluate. $$\sqrt{64}$$

Rewrite the radical expression in exponential notation. $$\sqrt[3]{h^{4}}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.