/*! 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 45 Simplify as completely as possib... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 45

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\sqrt{48}-\sqrt{75}$$

Short Answer

Expert verified
-\sqrt{3}

Step by step solution

01

Simplify the square roots

Simplify the square roots of 48 and 75 by finding their prime factorizations and identifying perfect squares.o \(48 = 16 \times 3 = 4^2 \times 3\rightarrow \ \sqrt{48} = \sqrt{4^2 \times 3} = 4\sqrt{3}\) \(75 = 25 \times 3 = 5^2 \times 3 \ \rightarrow\sqrt{75} = \sqrt{5^2 \times 3} = 5 \sqrt{3} \)
02

Substitute the simplified square roots back into the expression

Use the simplified forms of the square roots in the given expression.\(\sqrt{48} - \sqrt{75} \rightarrow 4\sqrt{3} - 5\sqrt{3}\)
03

Combine like terms

Combine the like terms (i.e., the square root terms with the same radicand).o\(4\sqrt{3} - 5\sqrt{3}= (4-5)\sqrt{3} = -\sqrt{3}\)

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.

Prime Factorization
Prime factorization is a method used to break down a number into its prime number components. These are the numbers that only have two divisors: 1 and themselves. For example, the number 48 can be broken down into prime factors by dividing by the smallest prime numbers successively. You get:
  • 48 梅 2 = 24
  • 24 梅 2 = 12
  • 12 梅 2 = 6
  • 6 梅 2 = 3
Here, 48 = 2 脳 2 脳 2 脳 2 脳 3 or written as powers of prime numbers, 48 = 2^4 脳 3.
Similarly, for 75, the smallest prime number factor is 3 and 5:
  • 75 梅 3 = 25
  • 25 梅 5 = 5
  • 5 梅 5 = 1
This gives us 75 = 3 脳 5 脳 5 or written as powers of prime numbers, 75 = 3 脳 5^2.
Prime factorization helps in simplifying square roots, as shown in the solution steps.
Perfect Squares
Perfect squares are numbers that are the square of integers. For example, 16 is a perfect square because it is equal to 4 脳 4 or 4^2. Identifying perfect squares is crucial when simplifying square roots since it allows us to express the square root in the simplest form.
In the exercise, we saw:
  • 48 = 16 脳 3 = 4^2 脳 3
So, \(\root{48} = \root{4^2 脳 3} = 4 脳 \root{3}\)
Similarly,
  • 75 = 25 脳 3 = 5^2 脳 3
Therefore, \(\root{75} = \root{5^2 脳 3} = 5 脳 \root{3}\). Using perfect squares makes it easier to simplify the expressions.
Combining Like Terms
Combining like terms is essential when simplifying expressions involving square roots or other algebraic terms. 'Like terms' are those that have the same variable components. For instance, in the expression \(4\root{3} - 5\root{3}, both terms have the same \root{3},\) so they are like terms.
In the exercise, after simplifying the square roots of 48 and 75:
  • \(\root{48} - \root{75} = 4\root{3} - 5\root{3}\)
To combine, we subtract the coefficients of the like terms:
  • \(4\root{3} - 5\root{3} = (4 - 5)\root{3} = -\root{3}\)
This step helps reduce the expression to its simplest form.

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 the method you think is the most appropriate to solve the given equation. Check your answers by using a different method. $$\frac{x}{x+1}=\frac{4}{x+4}$$

Multiply and simplify. $$(x-h)^{2}$$

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$(y+3)(y-5)=(2 y+5)(y-3)$$

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}-5 x-6$$

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$(2 a-0.3)^{2}=7.5$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.