/*! 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 75 Is the expression \(3 \sqrt{2}+\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 75

Is the expression \(3 \sqrt{2}+\sqrt{50}\) in simplest radical form? Defend your answer.

Short Answer

Expert verified
Yes, \(3\sqrt{2} + \sqrt{50} = 8\sqrt{2}\) is in simplest radical form.

Step by step solution

01

Understanding Simplest Radical Form

An expression is said to be in its simplest radical form when all the radicals are simplified, and no radicals appear in the denominator. To simplify a radical, the number inside the square root should be factored such that any perfect squares can be removed from under the radical.
02

Simplify \( \sqrt{50} \)

The number 50 can be factored into 25 and 2, where 25 is a perfect square. Thus, \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \).
03

Substitute and Combine Terms

Substitute \( \sqrt{50} \) with its simplified form in the expression. The expression becomes \( 3\sqrt{2} + 5\sqrt{2} \). Since both terms are like terms (both have \( \sqrt{2} \)), you can combine them: \( (3 + 5)\sqrt{2} = 8\sqrt{2} \).
04

Confirm the Simplest Radical Form

The expression \( 8\sqrt{2} \) is in its simplest radical form as 8 is an integer, and there are no further perfect square factors within the radical.

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.

Simplest Radical Form
Simplifying an expression into its simplest radical form means making sure the radicand鈥攖he number under the square root鈥攄oesn't contain any perfect square factors.
  • A perfect square is a number that can be expressed as the product of an integer with itself, such as 4, 9, 16, etc.
  • When an expression is in simplest radical form, all possible perfect squares are factored out, leaving only non-perfect square factors within the radical.
  • Additionally, the expression should not have radicals in the denominator.
For example, consider the expression \( \sqrt{50} \). The number 50 can be broken down as \( 25 \times 2 \), where 25 is a perfect square. Therefore, \( \sqrt{50} \) simplifies to \( \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \). This step ensures there are no perfect squares inside the radical.
Factoring Radicals
Factoring radicals involves expressing the number inside the radical as a product of its prime factors, simplifying whenever possible.
  • To simplify \( \sqrt{n} \), first find if \( n \) can be divided into factors, where one of them is a perfect square.
  • Identify the largest perfect square factor to simplify efficiently.
  • Extract the square root of the perfect square factor outside the radical. This reduces the complexity under the radical symbol.
In the exercise example, \( \sqrt{50} \) is decomposed into \( \sqrt{25 \times 2} \), then further simplified to \( 5 \sqrt{2} \). Understanding how to factor numbers into prime powers allows you to identify which radicals can be rewritten in simpler forms.
Combining Like Terms
Combining like terms is a crucial step in simplifying radical expressions. This process involves adding or subtracting terms that have the identical radicand part.
  • Like terms contain the same variable factor or literal part, which in the case of radicals means the same square root part.
  • Only coefficients are affected when combining like terms. The radical part remains unchanged.
In the expression \( 3\sqrt{2} + 5\sqrt{2} \), both terms have \( \sqrt{2} \) as their radicand, making them like terms. Thus, you combine them by adding their coefficients, resulting in \( 8\sqrt{2} \). This is how you effectively simplify the sum of radicals.

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

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((3.14)(10)^{10}\)

Use your calculator to evaluate each of the following. Express final answers in ordinary notation. (a) \((27,000)^{2}\) (b) \((450,000)^{2}\) (c) \((14,800)^{2}\) (d) \((1700)^{3}\)(e) \((900)^{4}\) (f) \((60)^{5}\) (g) \((0.0213)^{2}\) (h) \((0.000213)^{2}\) (i) \((0.000198)^{2}\) (j) \((0.000009)^{3}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{8}}{\sqrt[4]{4}}\)

Explain how you would evaluate \(27^{\frac{2}{3}}\) without a calculator.

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). \(500,000,000\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

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.