/*! 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 29 Simplify by factoring. $$\sqrt... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 29

Simplify by factoring. $$\sqrt{27}$$

Short Answer

Expert verified
\( \sqrt{27} = 3 \sqrt{3} \)

Step by step solution

01

- Identify Perfect Square Factors

Identify the perfect square factors of 27. The number 27 can be factored into 9 and 3, where 9 is a perfect square.
02

- Break Down the Square Root

Rewrite the square root of 27 as the product of the square roots of the factors: \( \sqrt{27} = \sqrt{9 \times 3} \).
03

- Simplify the Square Root

Use the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Therefore, \( \sqrt{27} = \sqrt{9} \times \sqrt{3} \).
04

- Evaluate the Perfect Square Root

Since 9 is a perfect square, \( \sqrt{9} = 3 \). Therefore, \( \sqrt{27} = 3 \times \sqrt{3} \).

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.

Factoring
Factoring is a critical concept in simplifying square roots. When you factor a number, you break it down into numbers that, when multiplied together, give you the original number.
Let's take the number 27 as an example. We need to identify factors of 27 that include a perfect square. Factors of 27 are 1, 3, 9, and 27.
  • Of these factors, 9 is a perfect square because it can be written as 32.
  • Thus, we can factor 27 as 9 脳 3.
Remember, factoring is useful in many areas of math, especially when dealing with square roots, because it helps simplify the expression further.
Perfect Square
A perfect square is a number that can be expressed as the product of an integer with itself. For instance, 9 is a perfect square because it equals 3 脳 3, or 32.
Why is it important to recognize perfect squares?
  • Perfect squares help simplify square roots easily.
  • In our example, we used 9 because it is a perfect square.
When simplifying square roots, finding a perfect square within the factors allows you to take the square root of that number easily. This makes the expression simpler and easier to handle in calculations.
Square Root Properties
Understanding the properties of square roots will help you manipulate and simplify them effectively. One key property is that the square root of a product equals the product of the square roots of each factor.
Mathematically, it means: \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] Using our example:
  • We first factor 27 as 9 脳 3.
  • This gives us \( \sqrt{27} = \sqrt{9 \times 3} \).
  • Using the property, we break it down to \( \sqrt{9} \times \sqrt{3} \).
Now, since \( \sqrt{9} \) is 3, the expression simplifies to 3\( \sqrt{3} \). Another property is that if you have the square root of a perfect square, it simplifies to an integer. Always use these properties to make your work easier and clearer.

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

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$\sqrt{6} \sqrt{3}$$

Describe a procedure that uses the distance formula to determine whether three points, \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right),\) are vertices of a right triangle.

Simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[4]{810 x^{9}} $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{s^{2} t^{4}} \sqrt[3]{s^{4} t^{6}} $$

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{a^{-1}+b^{-1}}{a b}[7.5] $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.