/*! 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 a. Write the first ten perfect-s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 3

a. Write the first ten perfect-square integers. b. Write the first ten perfect-cube integers.

Short Answer

Expert verified
The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The first ten perfect cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Step by step solution

01

Understanding Perfect Squares

A perfect square is an integer that is the square of an integer. This means it can be written in the form of \( n^2 \) where \( n \) is an integer.
02

Finding the First Ten Perfect Squares

To find the first ten perfect squares, we square the first ten positive integers, which are 1 through 10. \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), \(6^2 = 36\), \(7^2 = 49\), \(8^2 = 64\), \(9^2 = 81\), \(10^2 = 100\). Thus, the first ten perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
03

Understanding Perfect Cubes

A perfect cube is an integer that is the cube of an integer. This means it can be written in the form of \( n^3 \) where \( n \) is an integer.
04

Finding the First Ten Perfect Cubes

To find the first ten perfect cubes, we cube the first ten positive integers, which are 1 through 10. \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\), \(5^3 = 125\), \(6^3 = 216\), \(7^3 = 343\), \(8^3 = 512\), \(9^3 = 729\), \(10^3 = 1000\). Thus, the first ten perfect cubes are: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

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.

Perfect Squares
Perfect squares are numbers that can be expressed as the product of an integer with itself. In simpler terms, a perfect square results when you take a whole number and multiply it by itself.

For example:
  • When you square 5, you get 25 because 5 multiplied by 5 equals 25. Thus, 25 is a perfect square.
  • Similarly, 64 is a perfect square, as it equals 8 times 8.

The list of the first ten perfect squares starts with squaring the integer 1 up through squaring 10. You will have:
  • 1 (from 1脳1)
  • 4 (from 2脳2)
  • 9 (from 3脳3)
  • 16 (from 4脳4)
  • 25 (from 5脳5)
  • 36 (from 6脳6)
  • 49 (from 7脳7)
  • 64 (from 8脳8)
  • 81 (from 9脳9)
  • 100 (from 10脳10)
Recognizing perfect squares can help in many areas of math, such as simplifying expressions or solving quadratic equations.
Perfect Cubes
A perfect cube is a number that is the result of multiplying an integer by itself twice more, which means adding two more multiplications after the first.

Here鈥檚 an easy way to see it:
  • If you take the number 3 and multiply it three times as in 3脳3脳3, you get 27, making 27 a perfect cube.
  • Similarly, 125 is a perfect cube derived from 5脳5脳5.

To identify the first ten perfect cubes, you cube the numbers 1 through 10:
  • 1 (from 1脳1脳1)
  • 8 (from 2脳2脳2)
  • 27 (from 3脳3脳3)
  • 64 (from 4脳4脳4)
  • 125 (from 5脳5脳5)
  • 216 (from 6脳6脳6)
  • 343 (from 7脳7脳7)
  • 512 (from 8脳8脳8)
  • 729 (from 9脳9脳9)
  • 1000 (from 10脳10脳10)
Perfect cubes are useful when dealing with higher-level equations, particularly those involving exponents and roots.
Integer Relationships
Understanding the relationship between integers and their squares and cubes is foundational in mathematics.

Here are a few neat observations;
  • Each perfect square or cube comes from multiplying an integer by itself or including additional steps like itself twice more in the case of cubes.
  • Perfect squares grow quickly as you move to higher integers because each step multiplies by a larger number (e.g., squaring moves 4 to 16 from 2 to 4).
  • Perfect cubes grow even faster, since the multiplication is performed an extra time. For example, cubing 4 gives 64, which is already much larger than squaring it.

Developing a solid understanding of these relationships guides students through more complex math concepts, such as understanding geometric concepts and solving equations involving powers. Through practice, recognizing these basic integer relationships will enhance your ability to perform calculations and solve mathematical problems efficiently.

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

What polynomial should be subtracted from \(5 x^{3} y-5 x y+2 x\) to obtain the polynomial \(8 x^{3} y-7 x y+11 x ?\)

Explain the error. $$ \text { Factor: } \quad 4 g^{2}-16-(2 g+4)(2 g-4) $$

Find a polynomial equation of degree 3 with the solutions \(-3,-2,\) and 3

Factor each expression. $$ y^{2}-(2 x-t)^{2} $$

Factor each expression. $$ 1-y^{6} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.