/*! 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 13 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 13

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$\sqrt{72} \sqrt{\frac{5}{2}}$$

Short Answer

Expert verified
The simplest form is \(6\sqrt{5}\).

Step by step solution

01

Simplify each square root individually

First, we simplify each square root separately. \(\sqrt{72}\) can be broken down into \(\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\) because \(36\) is a perfect square. For \(\sqrt{\frac{5}{2}}\), we write it as \(\frac{\sqrt{5}}{\sqrt{2}}\).
02

Simplify the expression

Now, combine the results from the first step: \[ 6\sqrt{2} \times \frac{\sqrt{5}}{\sqrt{2}} = 6 \times \frac{\sqrt{2} \cdot \sqrt{5}}{\sqrt{2}} \]The \(\sqrt{2}\) in the numerator and denominator cancel each other out, resulting in \(6\sqrt{5}\).
03

Check for rationalized form of the denominator

Verify that the denominator is rationalized. In this case, the expression \(6\sqrt{5}\) does not have a denominator to rationalize, so it is already in simplest form.

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.

Simplifying Square Roots
Simplifying square roots involves breaking down a number or expression under a square root into its simplest form. The goal is to identify and extract perfect squares from the radicand (the number inside the square root).
To simplify
  • Identify perfect squares: Look for perfect square factors of the number under the square root. Perfect squares are numbers like 1, 4, 9, 16, 25, 36, etc.
  • Extract and simplify: Rewrite the square root as a product of two square roots, and simplify. For example, \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).
  • Leave non-perfect squares: Any number that is not a perfect square remains under the square root.
In our example, \(\sqrt{72}\) simplifies to \(6\sqrt{2}\), helping us to break down complex expressions into simpler parts.
Rationalizing Denominators
Rationalizing the denominator is the process of ensuring there are no square root terms in the denominator of a fraction.
To do this:
  • Multiply numerator and denominator by the same square root: This process involves multiplying the fraction by a clever form of one, such as \(\frac{\sqrt{b}}{\sqrt{b}}\), where \(b\) is the term in the denominator.
  • Simplify further: Often, this multiplication turns the denominator into a whole number, as square roots in the denominator cancel each other out.
For instance, given the expression \(\frac{\sqrt{5}}{\sqrt{2}}\), multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\) to get \(\frac{\sqrt{10}}{2}\). The denominator is now a rational number, 2, free from square roots.
Multiplying Square Roots
Multiplying square roots simplifies expressions involving multiple root terms. When multiplying square roots, you can combine them under a single root, provided they have the same radicand.
Here's how:
  • Use the associative and commutative properties: These properties allow us to multiply two square roots as a single operation \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b}\).
  • Combine and simplify: Multiply the numbers inside the square roots and simplify where possible.
  • Cancel common factors: If the expression involves simplification, like common terms in numerator and denominator, cancel them to further clean up your result.
In the example we examined, multiply \(6\sqrt{2} \) by \( \frac{\sqrt{5}}{\sqrt{2}} \), resulting in \(6 \cdot \sqrt{10}\), ensuring your final answer is in its simplest form. This showcases how multiplication helps consolidate square root terms efficiently.

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

Perform the indicated operations. $$\text { Is }\left[\left(4^{1 / 2}\right)^{-3 / 4}+\left(\frac{1}{4^{3}}\right)^{1 / 8}\right]^{4}=2 ?$$

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. $$\sqrt[3]{5-\sqrt{17}} \sqrt[3]{5+\sqrt{17}}$$

Use a calculator to evaluate each expression. $$(3 n-1)^{-2 / 3}(1-n)-(3 n-1)^{1 / 3}$$

Use a calculator to evaluate each expression. $$\frac{3^{-1} a^{1 / 2}}{4^{-1 / 2} b} \div \frac{9^{1 / 2} a^{-1 / 3}}{2 b^{-1 / 4}}$$

Use a calculator to evaluate each expression. $$\frac{s^{1 / 4} s^{2 / 3}}{5 s^{-1}}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

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.