/*! 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 18 Simplify each radical. $$\sqrt... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 18

Simplify each radical. $$\sqrt{18}$$

Short Answer

Expert verified
The simplified form of \(\sqrt{18}\) is \(3\sqrt{2}\)

Step by step solution

01

Prime Factorization

Firstly, let's perform a prime factorization for the number 18. For this, the number is broken down into the product of prime numbers. The prime factorisation of 18 is \(2 \times 3^2\).
02

Radical Simplification

The next step is to simplify the radical. This is achieved by taking pairs of identical prime numbers from under the root and moving one number from each pair outside of the root symbol. From the prime factorization, two prime numbers 3 could be moved outside the square root symbol. Resulting in \(3\sqrt{2}\).

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
Breaking down numbers into their prime factors is a key step when simplifying radicals. Prime factorization involves expressing a number as a product of its prime numbers. For instance, with the number 18, we can break it down into its prime components: 18 can be divided by 2, which is the smallest prime number, resulting in 9. Then, 9 can be further divided into 3 and 3, both prime numbers. So, the prime factorization of 18 is \( 2 \times 3^2 \).
This step allows us to work with smaller components of the number, which makes simplifying expressions involving radicals easier.
Radical Simplification
Simplifying radicals means reducing the expression to its simplest form. This involves using the prime factorization to identify perfect squares inside the radical.
For the square root of 18, which is written as \( \sqrt{18} \), we use the prime factorization \( 2 \times 3^2 \). Inside the radical, we have a pair of 3s. We can move one 3 outside the root, as \( \sqrt{3^2} = 3 \), leaving us with \( 3\sqrt{2} \).
In general, if a number inside the radical is a perfect square, it can be simplified by taking its square root and moving it outside.
Square Root Simplification
The square root simplification process is important for reducing complex expressions involving roots. It involves rewriting the square root in a way that makes calculations easier.
Given \( \sqrt{18} \), through prime factorization, you find \( 18 = 2 \times 3^2 \). In this breakdown, \( 3^2 \) is a perfect square. So, \( \sqrt{18} = \sqrt{3^2 \times 2} = 3\sqrt{2} \).
  • Look for pairs of identical numbers inside the radical.
  • Move each pair's root outside as a single number, simplifying the expression.
This kind of simplification transforms the expression into one that's easier to work with, especially for further calculations or comparisons.

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

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\frac{3}{4} \sqrt{192 a^{3} b^{5}}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$(-27)^{-4 / 3}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$-243^{3 / 5}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{1,600 x y^{2} z^{3}}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\left(3^{2 / 3} 5^{1 / 3}\right)^{3}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.