Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q11.

Expert-verified

Algebra 1

Found in: Page 10

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Name the set or sets of numbers to which each real number belongs.
$\sqrt{\frac{82}{20}}$

The real number $\sqrt{\frac{82}{20}}$ belongs to the set of irrational numbers

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step-1. Apply the concept of the real number system.

Real numbers include all the numbers except complex numbers and have the following five subsets:

Natural numbers: Includes counting objects and starting from 1.
Whole numbers: Includes the set of natural numbers along with 0.
Integers: Z = Includes numbers that are not fraction (positive and negative whole numbers)
Rational numbers: Includes the numbers which can be written in the form of $\frac{p}{q}$ where p and q are integers, $q \neq 0$ .
Irrational numbers: Includes numbers that cannot be written in the form of $\frac{p}{q}$ where p and q are integers, $q \neq 0$ .

Step-2. Examples of the real number system.

Natural numbers: $\{1, 2, 3, 4, . . .\}$
Whole numbers: $\{0, 1, 2, 3, 4, . . .\}$
Integers: Measurement of debts, temperatures, etc., fall under the set of integers $\{. . ., - 3, - 2, - 1, 0, 1, 2, 3, . . .\}$
Rational numbers: If we cut a cake into equal pieces, then we may have a piece that represents a fraction like $\frac{5}{6}, 1.5 (= \frac{3}{2}), \frac{6}{9}, \frac{8}{3}$
Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as $\sqrt{2}, \sqrt[3]{5}, - \sqrt{3}$

Step-3. Analyze the given real number.

Consider the given real number $\sqrt{\frac{82}{20}}$

We can rewrite the given real number as:

$\sqrt{\frac{82}{20}} = \sqrt{\frac{2 \times 41}{2 \times 10}} = \sqrt{\frac{41}{10}}$

Now we will check each subset of real numbers:

Natural numbers: They are positive, countable and start from 1. So, $\sqrt{\frac{82}{20}} = \sqrt{\frac{41}{10}}$ not being a counting object is not a natural number.
Whole numbers: They are natural numbers including 0. So, $\sqrt{\frac{82}{20}} = \sqrt{\frac{41}{10}}$ not being a natural number is not a whole number as well.
Integers: They are whole numbers that are both positive and negative. Since $\sqrt{\frac{82}{20}} = \sqrt{\frac{41}{10}}$ is not a whole number, it is not an integer as well.
Rational numbers: They can be written in the form $\frac{p}{q}$ where p and q are integers, $q \neq 0$ . Since $\sqrt{\frac{82}{20}}$ cannot be written in the exact form, it is not a rational number.
Irrational numbers: They cannot be written in the form $\frac{p}{q}$ where p and q are integers, $q \neq 0$ . Since from the above point, $\sqrt{\frac{82}{20}}$ cannot be written in the form $\frac{p}{q}$ , it is an irrational number.

Therefore, the real number $\sqrt{\frac{82}{20}}$ belongs to the set of irrational numbers.

Most popular questions for Math Textbooks

Which expression best represents the perimeter of the rectangle.

$\begin{array}{l} F) \frac{r^{4}}{9} \\ G) l + w \\ H) 2 l + 2 w \\ J) 4 (l + w) \end{array}$

Write an algebraic expression for each verbal expression.

one-third of a number

Evaluate each expression if $a = 4$ , $b = 6$ and $c = 8$ .

$\frac{b (9 - c)}{a^{2}}$

Evaluate the expression $3 + 42 \cdot 2 - 5$ .

Write each repeating decimal as a fraction in simplest form.

$0 . \bar{5}$

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free