/*! 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 56 In Exercises \(55-68,\) multiply... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 56

In Exercises \(55-68,\) multiply and, if possible, simplify. $$\sqrt{3} \cdot \sqrt{6}$$

Short Answer

Expert verified
The result after multiplication and simplification is \(3 \cdot \sqrt{2}\)

Step by step solution

01

Multiply the square roots

Start by multiplying the two square roots. This can be stated as \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \). So, \( \sqrt{3} \cdot \sqrt{6} = \sqrt{3 \cdot 6} \)
02

Evaluate the multiplication

Next, carry out the multiplication under the square root. \(3 \cdot 6 = 18\), so we have \( \sqrt{18} \)
03

Simplify the square root

Then, simplify the square root by identifying any perfect square factors of 18. We can write 18 as \(2 \cdot 3^2\), and the square root of \(3^2\) is 3. So, \( \sqrt{18} = \sqrt{2 \cdot 3^2} = 3 \cdot \sqrt{2} \)

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 Square Roots
Multiplying square roots might initially seem challenging, but it follows a specific property of radicals that states when you multiply two square roots, the result is the square root of the product of the two values. In algebraic terms, we express this as \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).

For instance, when multiplying \( \sqrt{3} \cdot \sqrt{6} \), we combine the two under a single radical, giving us \( \sqrt{3 \times 6} = \sqrt{18} \). The process promotes a streamlined approach to handling square roots in multiplication, avoiding complex procedures.
Radical Expressions
Radical expressions involve numbers under the symbol known as the radical sign (\( \sqrt{} \)). When dealing with square roots, the number under the radical is called the radicand. To work with these expressions effectively, it's essential to understand properties like the product and quotient rules for square roots.

Remember that multiplying and dividing radicals can be done only when they have the same index (the small number written just outside and above the radical sign, which is 2 for square roots). Also, it's necessary to simplify these expressions as much as possible to find the most straightforward form.
Perfect Square Factors
Identifying perfect square factors within a radical expression is crucial to simplification. A perfect square is the product of an integer multiplied by itself, such as \(4, 9, 16\), etc. When simplifying a radical like \( \sqrt{18} \), you search for perfect square factors of the radicand - in this case, \(18 = 2 \times 3^2\).

Since \(3^2\) is a perfect square (and thus its square root is an integer), it can be taken out from under the radical sign, simplifying the original expression to \(3 \cdot \sqrt{2}\). This step efficiently reduces the complexity of the radical expression.
Algebraic Simplification
Algebraic simplification encompasses a variety of techniques used to transform expressions into their simplest form. When focusing on radicals, this includes finding and extracting perfect square factors, as previously discussed. Always strive to express numbers in their simplest radical form, which may also involve rationalizing the denominator in fraction cases.

Simplification doesn't change the value of the expression; it only makes it cleaner and more manageable. It's a fundamental skill in algebra that aids in solving equations and facilitates understanding relationships between variables.

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 Exercises \(25-52,\) begin by simplifying the expression. Then rationalize the denominator using the simplified expression. $$\frac{\sqrt{27 x^{2}}}{\sqrt{12 y^{3}}}$$

Simplify: \(\sqrt{[6-(-4)]^{2}+[2-(-3)]^{2}}\).

Graph the solution set of the system: $$\left\\{\begin{aligned}-3 x+4 y & \leq 12 \\\x & \geq 2\end{aligned}\right.$$ (Section 4.5, Example 3)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$2^{\frac{1}{2}} \cdot 2^{\frac{3}{2}}=\left(\frac{1}{4}\right)^{-1}$$

In Exercises \(94-97,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 \sqrt{x}}{x \sqrt{6}}=\frac{\sqrt{6 x}}{2 x} \text { for } x>0$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.