/*! 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 182 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 182

In the following exercises, simplify. $$ \sqrt{72}-\sqrt{98} $$

Short Answer

Expert verified
-\(\sqrt{2}\).

Step by step solution

01

Simplify each square root

First, simplify the square root of each term. \[\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.\]Next, simplify \(\sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}\).
02

Substitute simplified terms

Now substitute the simplified forms into the original expression: \(6\sqrt{2} - 7\sqrt{2}\)
03

Combine like terms

Finally, combine like terms to get the simplified expression: \(6\sqrt{2} - 7\sqrt{2} = -1\sqrt{2} = -\sqrt{2}\).

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.

Square Root Simplification
To simplify square roots, you should decompose the number inside the square root into its factors, ideally including a perfect square.
For example, simplify \(\sqrt{72}\) by finding two numbers that multiply to 72, one of which should be a perfect square. In this case, 72 can be expressed as 36 \times 2. Since 36 is a perfect square, \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.\)
Similarly, \(\sqrt{98}\) can be broken into factors 49 \times 2. With 49 being a perfect square, you get \(\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}.\)
Breaking numbers into their perfect square factors simplifies square roots significantly.
Combining Like Terms
Combining like terms is a crucial skill in algebra. It involves simplifying an expression by adding or subtracting terms that have identical variables and exponents.
Take the simplified terms from our example: \(6\sqrt{2} - 7\sqrt{2}.\)
Since both terms contain \(\sqrt{2},\) they are like terms and can be combined.
You adjust the coefficients (numbers in front), so: \(6\sqrt{2} - 7\sqrt{2} = (6 - 7)\sqrt{2} = -1\sqrt{2}.\)
Therefore, our simplified expression is \(-\sqrt{2}.\)
Always double-check to ensure the terms you're combining have the same radical parts to avoid mistakes.
Radical Expressions
A radical expression includes a root, such as a square root, cube root, etc.
In our example, \(\sqrt{72} - \sqrt{98},\) we deal with square roots.
Simplify each term first, then combine them. Simplifying these expressions helps in performing operations like addition or subtraction on them.
For example, converting \(\sqrt{72}\) to \(6\sqrt{2}\) and \(\sqrt{98}\) to \7\sqrt{2}\ makes it easier to see the next step: \(6\sqrt{2} - 7\sqrt{2}.\)
Radicals can often be intimidating, but breaking them down step by step makes them much more manageable.
Regular practice with radical expressions can help solidify these skills, making them easier to work with over time.

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, simplify. (a) \(\sqrt[8]{k^{8}}\) (b) \(\sqrt[6]{p^{6}}\)

(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. $$ \frac{\sqrt{28}}{\sqrt{63}} $$

In the following exercises, simplify by rationalizing the denominator. $$ \frac{3}{5 \sqrt{8}} $$

In the following exercises, solve. (a) \(\sqrt{u-3}+3=u\) (b) \(\sqrt{x+1}-x+1=0\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.