/*! 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 7 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 7

Determine whether each statement is true for \(n=1,2,\) and 3. $$n^{2} < n^{3}$$

Short Answer

Expert verified
The statement is false for n=1 and true for n=2 and n=3.

Step by step solution

01

Evaluate the statement for n=1

To determine if the statement is true for given values of n, start with n = 1. Substitute n = 1 into the inequality: \[ n^2 < n^3 \] For n = 1, this becomes: \[ 1^2 < 1^3 \] Calculate the powers: \[ 1 < 1 \] This statement is false because 1 is not less than 1.
02

Evaluate the statement for n=2

Next, test the value n = 2. Substitute n = 2 into the inequality: \[ n^2 < n^3 \] For n = 2, this becomes: \[ 2^2 < 2^3 \] Calculate the powers: \[ 4 < 8 \] This statement is true because 4 is less than 8.
03

Evaluate the statement for n=3

Finally, test the value n = 3. Substitute n = 3 into the inequality: \[ n^2 < n^3 \] For n = 3, this becomes: \[ 3^2 < 3^3 \] Calculate the powers: \[ 9 < 27 \] This statement is true because 9 is less than 27.

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.

Understanding Evaluating Inequalities
Evaluating inequalities involves determining if a mathematical statement comparing two values is true. These comparisons can involve numbers, variables, and algebraic expressions. For example, in the inequality \(n^2 < n^3\), we compare the square of a number \(n\) with its cube. To evaluate it for specific values of \(n\), you substitute these values into both sides of the inequality and compute the results.
Here鈥檚 a quick guide to evaluate inequalities:
  • Identify the values you want to test.
  • Substitute each value into the inequality.
  • Calculate the expressions.
  • Compare the results to determine if the statement is true or false.
By following these steps, you can easily decide whether the inequality holds for any given value.
Exploring Exponentiation
Exponentiation is a fundamental operation in algebra that involves raising a base number to the power of an exponent. The expression \(n^x\) means multiplying \(n\) by itself \(x\) times. For example, \(n^2\) means \(n \times n\), and \(n^3\) means \(n \times n \times n\).
Important properties of exponentiation include:
  • If the exponent is 1, the result is the base itself (\(n^1 = n\)).
  • Any number raised to the power of 0 is 1 (\(n^0 = 1\)).
  • The exponentiation operation is associative and not commutative, meaning \((a^b)^c = a^{bc}\) but not always \(a^b \times a^c = a^{b+c}\).
Understanding these properties is crucial for solving many algebraic problems. For instance, in our inequality \(n^2 < n^3\), knowing that any real number greater than 1 satisfies this relationship helps in evaluating the inequality.
Simplifying Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division). Simplifying these expressions helps make them easier to work with. Here鈥檚 how you simplify algebraic expressions:
  • Combine like terms (terms that have the same variable raised to the same power).
  • Use the distributive property to eliminate parentheses (\(a(b + c) = ab + ac\)).
  • Factor expressions when possible to find common factors.
For instance, in evaluating the inequality \(n^2 < n^3\) for specific values of \(n\), each side of the inequality is already in a simplified form, making it straightforward to plug in the values and compare the results. Simplifying algebraic expressions not only aids in solving inequalities but also plays a significant role in various areas of mathematics, making complex problems more manageable.

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

Use the addition rule to solve each problem. If \(P(A)=0.2, P(B)=0.3,\) and \(P(A \cap B)=0,\) then what is \(P(A \cup B) ?\)

Write the first three terms of each binomial expansion. $$(2 s-0.5 t)^{8}$$

Consider the sample space of 36 equally likely outcomes to the experiment in which a pair of dice is rolled. In each case determine whether the events \(A\) and \(B\) are mutually exclusive. \(A:\) One of the numbers is two. \(B:\) The sum is greater than nine.

Solve each problem. Electing Jones If the odds against Jones winning the election are 3 to \(5,\) then what is the probability that Jones will win the election?

Solve each problem. Tax Time If the probability of a tax return not being audited by the IRS is 0.91 , then what is the probability of a tax return being audited?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.