/*! 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 3 Identify each number as a perfec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 3

Identify each number as a perfect square or not a perfect square. $$-8$$

Short Answer

Expert verified
-8 is not a perfect square.

Step by step solution

01

Understanding perfect squares

A perfect square is an integer that is the square of an integer. For instance, numbers like 4 \( (=2^2) \), 9 \( (=3^2) \), and 16 \( (=4^2) \) are perfect squares.
02

Check if the number can be a perfect square

Before applying any operation, it's important to note that perfect squares are non-negative. This means a negative number, such as -8, can never be a perfect square.

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.

Integers
An integer is a whole number that can be positive, negative, or zero. Unlike fractions and decimals, integers do not have a fractional or decimal component.
Here are some key characteristics of integers:
  • Integers include numbers like -3, 0, and 7.
  • The set of integers is infinite, meaning there is no largest or smallest integer.
  • Every integer is evenly divisible by itself and 1.
Integer properties play a crucial role in determining if a number is a perfect square. To be a perfect square, an integer must be the square of another integer, e.g., 9 which is the square of 3. Understanding integers is foundational when studying perfect squares, as it limits potential candidates to non-fractional and non-decimal numbers.
Non-negative Numbers
Non-negative numbers are all numbers greater than or equal to zero. They include all the positive integers and zero but do not include any negative numbers. This is a key factor in determining perfect squares.
  • Examples of non-negative numbers include 0, 1, 2, and so on.
  • Non-negative numbers are useful in measuring quantities that cannot be negative, like distance or time.
  • When considering perfect squares, only non-negative numbers qualify, as squaring any real number results in a non-negative value.
Understanding that perfect squares can only be non-negative helps quickly rule out any negative integers, such as -8, from being considered as perfect squares.
Square of an Integer
The square of an integer is found by multiplying the integer by itself. This mathematical operation results in a value called a perfect square if the operation began with an integer.
  • The notation often used to express this is: if \( x \) is an integer, then the square is \( x^2 \).
  • Common perfect squares include \( 0^2 = 0 \), \( 1^2 = 1 \), \( 2^2 = 4 \), and so on.
  • It is important to note that negative numbers, when squared, become positive. Thus, a negative number cannot directly be a perfect square.
This knowledge underpins the determination of perfect squares. Since squaring a negative integer results in a positive value, negatives like -8 cannot be perfect squares. Perfect squares emerge from both positive integers and zero.

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

How do you know when a radical has been simplified?

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$243^{2 / 5}$$

When is \(\sqrt{x^{2}} \neq x ?\)

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\frac{a^{2 / 5} a^{1 / 5}}{a^{-1 / 3}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{54 x^{3} y^{4} z^{6}}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.