/*! 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 15 Simplify each radical. $$ \s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 15

Simplify each radical. $$ \sqrt{45} $$

Short Answer

Expert verified
The simplified form of \(\sqrt{45}\) is \(3\sqrt{5}\).

Step by step solution

01

Identify Perfect Squares

Find the largest perfect square that is a factor of 45. The perfect squares less than 45 are 1, 4, 9, 16, 25, and 36. The largest perfect square factor of 45 is 9 because 45 is equal to 9 multiplied by 5.
02

Rewrite the Expression

Rewrite the radical expression by separating it into the product of the square root of the perfect square and the square root of the remaining factor. Thus, \[ \ \text { } \ \text { } \ \ \sqrt{45} = \sqrt{9} \cdot \sqrt{5} \]
03

Simplify the Expression

Simplify the square root of the perfect square. Since the square root of 9 is 3, \[\sqrt{9} = 3.\] So, \[\sqrt{45} = 3 \cdot \sqrt{5}.\]

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.

perfect square
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, 25, and 36 are all perfect squares. They are the results of squaring 1, 2, 3, 4, 5, and 6 respectively.
Perfect squares are important in simplifying radical expressions. By identifying the largest perfect square factor of a number, you can simplify the radical. For example, with the number 45, the largest perfect square factor is 9, because 45 = 9 脳 5.
Breaking down the number into its prime factors can help identify perfect square factors.
  • For 45, the prime factors are 3 脳 3 脳 5.
  • The largest perfect square factor is 3 脳 3 = 9.
Using perfect squares in this way simplifies the radical expression effectively.
radical expression
A radical expression includes a number inside a root symbol, like \( \sqrt{45} \). It's often the case that a radical can be simplified. For example, \( \sqrt{45} \) can be split into the product of \( \sqrt{9} \) and \( \sqrt{5} \).
Simplifying a radical means rewriting it in its simplest form. Most radicals aren't perfect squares, but many have perfect square factors. Recognizing these factors and breaking down the radical can make it simpler.
For example, to simplify \( \sqrt{45} \):
  • Identify the perfect square factor (9).
  • Separate the expression into two radicals: \( \sqrt{9} \cdot \sqrt{5} \).
  • Simplify the perfect square radical: \( \sqrt{9} = 3 \).
This results in \( 3 \sqrt{5} \), a simplified radical expression that's easier to understand and work with.
factorization
Factorization involves breaking down a number into its prime factors. It's an essential step in simplifying radicals because it helps identify perfect square factors.
For the number 45, the factorization process involves:
  • Finding the prime factors: 45 = 3 脳 3 脳 5.
  • Grouping the factors into a perfect square if possible: In this case, 3 脳 3 = 9 is a perfect square.

    • This leads to a simpler form of the radical. After factorizing, you rewrite the expression:
    \[ \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} \]
    Then simplify further: \( \sqrt{9} = 3 \) results in: \ \sqrt{45} = 3 \sqrt{5}, \ making it easier to work with the radical expression.

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

Solve each equation. (Hint: In Exercises 67 and 68, extend the concepts to fourth root radicals.) $$ \sqrt[4]{x^{2}+24 x}=3 $$

Find each product and simplify. $$ 2 \sqrt{10} \cdot 3 \sqrt{2} $$

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{900 y^{8}} $$

Simplify each radical. $$ \sqrt[3]{\frac{m^{12}}{8}} $$

Solve each problem. Three times the square root of 2 equals the square root of the sum of some number and \(10 .\) Find the number.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.