/*! 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 244 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 244

In the following exercises, simplify. $$ (-2 \sqrt{11})(-4 \sqrt{22}) $$

Short Answer

Expert verified
88 \sqrt{2}

Step by step solution

01

- Multiply the Constants

First, multiply the constants in the expression. The constants are -2 and -4. $$-2 \times -4 = 8$$
02

- Multiply the Radicals

Next, multiply the radicals together. The radicals are \( \sqrt{11} \) and \( \sqrt{22} \). Using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), we get: $$ \sqrt{11} \cdot \sqrt{22} = \sqrt{242} $$
03

- Combine the Results

Now combine the product of the constants and the radicals: $$ 8 \sqrt{242} $$
04

- Simplify the Radical (if possible)

Finally, check if \( \sqrt{242} \) can be simplified further. The prime factors of 242 are 2, 11, and 11. Thus \( 242 = 2 \cdot 11^2 \).Hence, \( \sqrt{242} = \sqrt{2 \cdot 11^2} = 11 \sqrt{2} \). So the simplified form is: $$ 8 \cdot 11 \sqrt{2} = 88 \sqrt{2} $$

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.

Multiplying Constants
Multiplying constants is an essential step when working with expressions that include numbers and radicals. Constants are the numbers written without a radical sign. For example, in the expression \( -2 \sqrt{11} \), \(-2\) is the constant part. To multiply constants, follow these simple steps:
  • Identify the constants in the expression.
  • Multiply them directly just as you would with regular numbers.
So for our example, we take \(-2\) and \(-4\). Using multiplication rules:
\(-2 \times -4 = 8\).
Notice that multiplying two negative numbers gives a positive result. This is always the case with negative multiplication.
Multiplying Radicals
After dealing with the constants, we move on to multiplying the radicals. Radicals are the parts of the expression under the root sign, like \( \sqrt{11} \) and \(\b\sqrt{22} \). To multiply radicals:
  • Combine the numbers under the radicals using multiplication.
  • Use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
For example, with \( \sqrt{11} \) and \(\b\sqrt{22} \), we combine them as:
\(\b\sqrt{11} \cdot \sqrt{22} = \sqrt{242} \).
This property allows us to combine the terms easily under a single radical.
Simplifying Square Roots
Simplifying square roots can help us make the final expression much neater. To simplify a square root:
  • Factor the number under the radical into its prime factors.
  • Look for pairs of prime factors.
  • Move pairs of prime factors out from under the radical, as they combine into a single number outside the radical.
In the given example, we need to simplify \( \sqrt{242} \). First, find its prime factors:
\(\b\sqrt{242} = \sqrt{2 \cdot 11^2} \).
Since \( 11^2 \) is a pair, it can be moved outside the radical:
\(\b\sqrt{242} = 11 \sqrt{2} \).
Finally, combine it with the constant we calculated earlier to get the simplified expression: \( 8 \cdot 11 \sqrt{2} = 88 \b\sqrt{2} \).

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

In the following exercises, solve. $$ 3 \sqrt{2 x-3}-20=7 $$

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{7}{40}} $$

In the following exercises, simplify and rationalize the denominator. $$ \frac{10}{\sqrt{11}} $$

(a) Simplify \(\sqrt{\frac{27}{3}}\) and explain all your steps. (b) Simplify \(\sqrt{\frac{27}{5}}\) and explain all your steps. (c) Why are the two methods of simplifying square roots different?

In the following exercises, simplify and rationalize the denominator. $$ \frac{8}{\sqrt{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

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.