/*! 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 79 Simplify. $$\frac{5+\sqrt{75}}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 79

Simplify. $$\frac{5+\sqrt{75}}{10}$$

Short Answer

Expert verified
\[ \frac{1 + \root{3}}{2} \]

Step by step solution

01

- Simplify the Radical

Start by simplifying the square root part of the expression. Notice that \(\root{75}\) can be simplified because 75 is 25 times 3. So, \(\root{75} = \root{25 \times 3}\). Since \(\root{25} = 5\), we have \(\root{75} = 5\root{3}\).
02

- Substitute Simplified Radical

Replace \( \root{75} \) in the original expression with its simplified form. This gives us: \[ \frac{5 + 5 \root{3}}{10} \]
03

- Factor Out Common Term

Factor out the common term 5 from the numerator. The expression becomes \[ \frac{5 (1 + \root{3})}{10} \]
04

- Simplify the Fraction

Simplify the fraction by dividing the numerator and the denominator by 5. This leaves us with: \[ \frac{1 + \root{3}}{2} \]

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.

Radical Simplification
When working with radicals, you should always look to simplify them. Start by breaking down the number inside the square root into its prime factors or factors that are perfect squares. For the number 75, we notice that it can be expressed as the product of 25 and 3. Why? Because 75 is equal to 25 times 3. Since 25 is a perfect square, this is helpful. As a result, we can express the square root of 75 as \(\root{75} = \root{25 \times 3}\). From here, we know that \(\root{25} = 5\), so it follows that \(\root{75} = 5\root{3}\). By simplifying the radical, you make the expression easier to work with in subsequent steps.
Factoring
Next, let's move to factoring. Factoring involves taking out common terms from an expression to simplify it. In the given problem, once we substituted \(\root{75}\) with \(\root{75} = 5\root{3}\) from the first step, the expression changed to \(\frac{5 + 5\root{3}}{10}\). Here, we notice that the numerator has a common factor of 5. We factor out 5 from the numerator, which transforms our expression into \(\frac{5 (1 + \root{3})}{10}\). Factoring is a crucial step as it reveals common factors that can be canceled out, making the simplification process smoother.
Fraction Simplification
The final step in our solution is fraction simplification. This process involves canceling common factors between the numerator and the denominator. From our last result, \(\frac{5 (1 + \root{3})}{10}\), we can see that both the 5 in the numerator and the 10 in the denominator have a common factor of 5. When we divide both by 5, the expression simplifies to \(\frac{1 + \root{3}}{2}\). Simple steps like these ensure that the fraction is in its simplest form, making it easier to understand and work with. Remember, removing common factors makes the overall fraction less complex and more manageable.

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

Use a calculator to find approximate solutions to the following equations. Round your answers to three decimal places. $$\sqrt[7]{x-5}=3.7$$

Simplify. $$\frac{\sqrt{x}}{\sqrt{x}+2}+\frac{3 \sqrt{x}}{\sqrt{x}-2}$$

Simplify each expression. $$5 \sqrt{w^{2} y}-7 \sqrt{w^{2} y}+6 \sqrt{w^{2} y}$$

Solve each equation. $$\sqrt{5 x^{2}+4 x+1}-x=0$$

Simplify. $$(-2 \sqrt{x y^{2} z})^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.