/*! 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 21 Simplify. $$ \sqrt{50} $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 21

Simplify. $$ \sqrt{50} $$

Short Answer

Expert verified
\( \scriptstyle \sqrt{50} = 5\sqrt{2} \)

Step by step solution

01

- Factor the radicand

First, factor the number inside the square root, which is 50. Find the prime factors of 50. We have: 50 = 2 脳 5 脳 5.
02

- Group the factors

Group the factors inside the square root such that one group can form a perfect square. For 50, it can be rewritten as \(\text{50} = 2 \times 5^2\).
03

- Simplify the square root

Simplify \(\text{50} = \text{2} \times 5^2\) to separate the perfect square. Take the square root of the perfect square (25) and leave the square root of the other number (2). Thus, we get \(\text{\footnotesize \sqrt{50} = \sqrt{2 \times 5^2} = 5 \sqrt{2}\).

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.

Prime Factorization
Prime factorization is a method used to break down a number into its simplest prime number components. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves, such as 2, 3, 5, and 7.
To find the prime factorization of 50, we start by dividing it by the smallest prime number, which is 2:
  • 50 divided by 2 is 25
  • Next, we continue with 25
  • 25 can be factored further into 5 and 5
Now, we have:
  • 50 = 2 脳 5 脳 5
This means that all factors of 50 are prime, and this process helps us in simplifying square roots.
Perfect Square
A perfect square is a number that is the square of an integer. For instance, 9 is a perfect square because it is 3^2, and 25 is a perfect square because it is 5^2.
Why do we need perfect squares? When simplifying square roots, it's easier to deal with numbers that are perfect squares because we know their exact square roots.
In our exercise, the number 50 can be broken down into the product of 2 and 25, where 25 is a perfect square since it can be written as 5^2.
By identifying perfect squares within the factors of our radicand (the number inside the square root), simplification becomes straightforward.
Simplifying Radicals
Simplifying radicals involves reducing the expression under the square root sign to its simplest form. This can be achieved by using prime factorization and identifying perfect squares.
Here's how it works for \(\text{\sqrt{50}}\):
  • First, we write 50 as the product of its prime factors: 2 脳 5^2.
  • Next, we separate the perfect square in the factorization: 2 脳 (5^2).
  • Then, we take the square root of the perfect square 5^2, which is 5.
  • The expression inside the square root now looks like this: \(\text{\sqrt{2} 脳 \sqrt{5^2}} = \sqrt{2} 脳 5\).
  • The square root of 50 is simplified to: 5 \(\text{\sqrt{2}}\).
By following these steps, you can simplify radicals and make complex expressions easier to handle.

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

Santiago drew a picture using line segments on a coordinate grid. He then multiplied the coordinates of all the endpoints by 1.5, plotted the resulting points on a new grid, and connected them to form a new picture. a. One segment in Santiago鈥檚 original drawing was 2 in. long. How long was the corresponding segment in the new drawing? b. One segment in the new drawing was 2 in. long. How long was the corresponding segment in Santiago鈥檚 original drawing?

Find the value of c in each equation. $$\sqrt[c]{8}=2$$

Consider this set of data. \(\begin{array}{lllll}{14.5} & {15.6} & {18.1} & {16.2} & {15.9}\end{array}\) a. Find the mean and the median of this data set. b. What two values can you add to the data set so that the median remains the same but the mean is higher? c. What two values can you add to the original data set so that the mean remains the same but the median is higher?

Find the value of \(t\) in each equation. \(3^{t}=729\)

Perspective drawings look three-dimensional. The projection method for making scale drawings is related to a method for making perspective drawings. On your own paper, follow the steps below to make a perspective drawing of a box. Use a pencil. a. Start by drawing a rectangle. This will be the front of your box. b. Choose a point outside your rectangle. This point is called the vanishing point for your drawing. Connect each vertex to that point, and then find the midpoint of each connecting segment. c. Connect the four midpoints you found in Part b to each other, in order. This gives you the back of the box. Then erase the lines connecting them to the vanishing point. d. To make the box clearer, erase the lines that should be hidden on the back of the box, or make them dashed. e. Follow the same steps to make a perspective drawing of a triangular prism. That is, start with a triangle (instead of a rectangle) and follow Part a鈥揹. f. In this method of three-dimensional drawing, at what step do you create a pair of similar figures? Explain.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.