/*! 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 87 Using \(\sqrt{50}\), explain how... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 87

Using \(\sqrt{50}\), explain how to simplify a square root.

Short Answer

Expert verified
The simplified form of \(\sqrt{50}\) is \(5*\sqrt{2}\).

Step by step solution

01

Understanding square roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3*3 is 9.
02

Factorize 50

Divide 50 into its prime factors. The prime factors of 50 are 2 and 5, and in factorized form, 50 equals to \(2*5*5\) or \(2*5^2\).
03

Simplify the square root

The square root can be simplified by applying the rule that the square root of the product equals the product of the square roots. Thus, \(\sqrt{50}\) equals to \(\sqrt{2*5^2}\), which simplifies to \(5*\sqrt{2}\).

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.

Square Root Concepts
Square roots are based on the idea of finding a number which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3, because multiplying 3 by itself gives you 9. This concept is central in mathematics, especially in geometry and algebra, to understand not just numbers but also relationships and equations.
\(\sqrt{50}\) illustrates this notion. It represents the non-negative root of 50. When simplifying square roots, identifying perfect squares within the number is crucial, as they allow for simplification. Keep in mind:
  • A perfect square is a number that can easily come from squaring another integer, like 1, 4, 9, 16, etc.
  • When a square root simplifies to a whole number, it indicates it's perfect. But, like \(\sqrt{50}\), if it can't perfectly resolve to a whole number, it's often reducible with other techniques.
Prime Factorization
Prime factorization involves breaking down a number into its prime constituent factors. Prime factors are numbers that cannot be divided further by anything except 1 and themselves, such as 2, 3, 5, 7, etc.
By practicing prime factorization on 50, we aim to simplify its square root. The factorization process begins by dividing 50 by the smallest prime number that fits neatly into it, which is 2 in this case.
  • The result, 25, further splits into two 5s because 25 is the result of \(5 * 5\).
  • Therefore, the entire set of prime factors for 50 is \(2 * 5^2\).
  • This decomposition aids in identifying parts of the square root that can simplify, making it manageable to work with algebraically.
Algebraic Simplification
Simplifying square roots through algebraic techniques helps maintain readability and usability of expressions. To simplify \(\sqrt{50}\), utilize the product property of square roots that states \(\sqrt{a*b} = \sqrt{a} * \sqrt{b}\).
Applying this to \(\sqrt{50}\):
  • Rewrite as \(\sqrt{2 * 5^2}\).
  • Split into \(\sqrt{2}\) and \(\sqrt{5^2}\).
  • Since squaring and square roots are inverse operations, \(\sqrt{5^2}\) simplifies to 5.
  • This leaves us with the expression 5\(\sqrt{2}\).
This simple form of \(\sqrt{50}\) is more concise and preferred in algebraic work, representing the conversion of complex calculations into easier forms without losing value.

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

Determine whether each sequence in Exercises 99-114 is arithmetic or geometric. Then find the next two terms. \(2,6,10,14, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2000, r=-1\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{20}\), when \(a_{1}=2, r=3\).

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=4, r=-3\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.