/*! 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 81 Which of the following are ratio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 81

Which of the following are rational and which are irrational? (a) \(-\sqrt{9}\) (b) \(0.375\) (c) \((3 \sqrt{2})(5 \sqrt{2})\) (d) \((1+\sqrt{3})^{2}\)

Short Answer

Expert verified
(a) Rational, (b) Rational, (c) Rational, (d) Irrational.

Step by step solution

01

Determine if (a) is Rational or Irrational

The square root of 9 is 3, so \(-\sqrt{9} = -3\). A rational number is any number that can be expressed as a quotient of two integers. Since -3 can be written as \(-\frac{3}{1}\), it is a rational number.
02

Determine if (b) is Rational or Irrational

The number 0.375 is a terminating decimal. Terminating decimals can be expressed as a fraction, \(\frac{375}{1000}\), which can be simplified to \(\frac{3}{8}\). Therefore, 0.375 is a rational number.
03

Determine if (c) is Rational or Irrational

Multiply the terms: \(3 \sqrt{2} \times 5 \sqrt{2} = 15 \times 2 = 30\). Since 30 can be written as \(\frac{30}{1}\), it is a rational number.
04

Determine if (d) is Rational or Irrational

Expand the expression \(1+\sqrt{3})^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\): \(= (1)^2 + 2(1)(\sqrt{3}) + (\sqrt{3})^2\), giving \(1 + 2\sqrt{3} + 3\), which equals \(4 + 2\sqrt{3}\). Because the term \(2\sqrt{3}\) is irrational, the entire expression is irrational.

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 Roots
Square roots play a fundamental role in mathematics. They ask the simple question: "Which number, when multiplied by itself, gives the original number?" For example, the square root of 9 is 3, because 3 脳 3 = 9. This concept is essential to determine if numbers like the \(-\sqrt{9}\) are rational or irrational.
When working with square roots, we often encounter perfect squares. Those are numbers like 1, 4, 9, 16, etc., whose square roots are integers.
  • If the square root results in an integer, the original number is usually a perfect square.
  • Square roots of non-perfect squares are usually irrational numbers because they don't have a simple fraction equivalent.
Understanding square roots helps in correctly identifying the nature of a number in any calculation.
Terminating Decimals
Terminating decimals are decimals that have a finite number of digits after the decimal point. An example found in the exercise is 0.375. It is easy to understand why they are considered rational numbers.
  • These decimals can always be expressed as fractions. For example, 0.375 converts to the fraction \(\frac{375}{1000}\), which further simplifies to \(\frac{3}{8}\).
  • Because they end after a certain number of decimal places, it is straightforward to convert them into a fraction using powers of 10.
When you know a decimal is terminating, you can quickly determine it as a rational number because it can be precisely expressed as a fraction.
Irrational Numbers
Irrational numbers cannot be written as simple fractions. They are numbers that have non-repeating, non-terminating decimal expansions.
An example from the exercise is the expression \(1+\sqrt{3})^2\). Expanding this results in a mix of whole numbers and irrational components like 2\(\sqrt{3}\), ensuring the entire expression stays irrational.
  • Irrational numbers include quantities like \(\pi\) and \(e\), as well as square roots of non-perfect squares.
  • They often appear in expressions involving roots of numbers that aren't square integers, resulting in a decimal that goes on forever without forming a repeatable pattern.
Understanding when a number cannot be expressed as a fraction is crucial for identifying irrational numbers.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction where both the numerator and denominator are integers. This means they can either be a whole number, a terminating decimal, or a repeating decimal.
  • For example, \(-3\) from \(-\sqrt{9}\) and 30 are rational because they can be written as \(-\frac{3}{1}\) and \(\frac{30}{1}\).
  • Any decimal that ends or repeats is also rational, as it can be converted to a fractional form.
Rational numbers are everywhere in daily life, from simple fractions like halves and quarters to prices and measurements.

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

Write an equation for the line through \((3,-3)\) that is (a) parallel to the line \(y=2 x+5\); (b) perpendicular to the line \(y=2 x+5\); (c) parallel to the line \(2 x+3 y=6\); (d) perpendicular to the line \(2 x+3 y=6\); (e) parallel to the line through \((-1,2)\) and \((3,-1)\); (f) parallel to the line \(x=8\); (g) perpendicular to the line \(x=8\).

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=3 \cos \left(x-\frac{\pi}{2}\right)-1 $$

Calculate. (a) \(\frac{234.1 \sin 1.56}{\cos 0.34}\) (b) \(\sin ^{2} 2.51+\sqrt{\cos 0.51}\)

Use the addition identity for the tangent to show that \(\tan (t+\pi)=\tan t\) for all \(t\) in the domain of \(\tan t\).

Verify the following are identities. (a) \(\frac{\sin u}{\csc u}+\frac{\cos u}{\sec u}=1\) (b) \(\left(1-\cos ^{2} x\right)\left(1+\cot ^{2} x\right)=1\) (c) \(\sin t(\csc t-\sin t)=\cos ^{2} t\) (d) \(\frac{1-\csc ^{2} t}{\csc ^{2} t}=\frac{-1}{\sec ^{2} t}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Probability and 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.