/*! 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 61 Simplify the radical expression.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 61

Simplify the radical expression. $$\sqrt{80}$$

Short Answer

Expert verified
The simplified form of \( \sqrt{80} \) is \( 4\sqrt{5} \)

Step by step solution

01

Prime Factorization of 80

First, find out the prime factors of 80. The prime factors of 80 are \(2^4 \times 5^1\). So, write the given expression as \( \sqrt {2^4 \times 5} \)
02

Apply Properties of Radicals

Now, applying the properties of radicals, split the radical into two separate radicals: \( \sqrt{2^4} \times \sqrt{5} \)
03

Evaluate and Simplify

Solving, \( \sqrt{2^4} = 2^2 = 4 \) and the radical \( \sqrt{5} \) cannot be simplified further. Therefore, the simplified expression becomes \( 4\sqrt{5} \).

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 process used to break down a composite number into its prime factors, which are the building blocks of all numbers. Every number greater than one is either a prime number itself or can be factored into prime numbers.
To factor the number 80, we begin by identifying the smallest prime number, which is 2, and determine if it divides into 80 evenly. Since 80 is even, it is divisible by 2:
  • 80 divided by 2 equals 40
  • 40 divided by 2 equals 20
  • 20 divided by 2 equals 10
  • 10 divided by 2 equals 5
At this point, the number 5 is a prime number, so the prime factorization of 80 is expressed as \(2^4 \times 5^1\). This decomposition is crucial for simplifying radical expressions by grouping factors into perfect squares.
Properties of Radicals
Radicals have unique properties that enable us to manipulate expressions efficiently when calculating square roots. One important property of radicals states that the square root of a product is equal to the product of the square roots:
  • \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
Applying this to the expression \(\sqrt{2^4 \times 5}\), we can split the radical into two parts: \(\sqrt{2^4} \times \sqrt{5}\).
By separating the factors, each part is evaluated individually, simplifying the process as much as possible.
Since some radicals are more straightforward to handle on their own, understanding and using this property helps make complex radical expressions easier to work with.
Square Root Simplification
Square root simplification involves reducing a square root expression to its simplest form, which makes mathematical calculations easier. The goal is to express the square root in such a way that no perfect square factors remain under the radical.
Given \(\sqrt{2^4 \times 5}\), simplify by first evaluating \(\sqrt{2^4}\). This is because 2 raised to the fourth power becomes a perfect square:
  • \(2^4 = 16\), and the square root of 16 is 4
Thus, \(\sqrt{2^4} = 4\). We are left with \(\sqrt{5}\), which cannot be simplified further since 5 is a prime number.
Therefore, combining these simplifications results in \(4\sqrt{5}\), making calculations involving this expression much clearer and more efficient.

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

You are making a 350 -mile car trip. You decide to drive a little faster to save time. Choose an expression for the time saved if the car's average speed \(s\) is increased by 5 miles per hour. $$\begin{array}{lllll} \text { (A) } \frac{350}{s+5} & \text { (B) } \frac{s+5}{350}-\frac{s}{350} & \text { (C) } \frac{350}{s}-\frac{350}{s+5} & \text { (D } 350(s+5)-350 s \end{array}$$

Completely factor the expression. $$15 x^{4}-50 x^{3}-40 x^{2}$$

Simplify the expression. $$\frac{x}{x-10}+\frac{x+4}{x+6}$$

Evaluate the expression. $$2^{4} \cdot 2^{3}$$

Simplify the expression. $$\frac{9}{x^{2}-3 x}+\frac{3}{x-3}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

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.