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Chapter 8: Problem 3

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

Short Answer

Expert verified
a. First ten perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. b. First ten perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Step by step solution

01

Define Perfect Square Integers

A perfect square integer is an integer that can be expressed as the square of another integer. For example, the perfect square of 3 is 3^2 = 9. To find the first ten perfect-square integers, we square the first ten natural numbers: 1^2, 2^2, 3^2, ..., 10^2.
02

Calculate Perfect Squares

Calculate the squares of the first ten natural numbers: 1. 1^2 = 1 2. 2^2 = 4 3. 3^2 = 9 4. 4^2 = 16 5. 5^2 = 25 6. 6^2 = 36 7. 7^2 = 49 8. 8^2 = 64 9. 9^2 = 81 10. 10^2 = 100. Thus, the first ten perfect-square integers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
03

Define Perfect Cube Integers

A perfect cube integer is an integer that can be expressed as the cube of another integer. For example, the perfect cube of 2 is 2^3 = 8. To find the first ten perfect-cube integers, we cube the first ten natural numbers: 1^3, 2^3, 3^3, ..., 10^3.
04

Calculate Perfect Cubes

Calculate the cubes of the first ten natural numbers: 1. 1^3 = 1 2. 2^3 = 8 3. 3^3 = 27 4. 4^3 = 64 5. 5^3 = 125 6. 6^3 = 216 7. 7^3 = 343 8. 8^3 = 512 9. 9^3 = 729 10. 10^3 = 1000. Thus, the first ten perfect-cube integers are: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Cubes
A perfect cube is an integer that results from multiplying an integer by itself twice. In other words, it is the cube of an integer. For example, the perfect cube of 2 is calculated as \(2 \times 2 \times 2 = 2^3 = 8\). Understanding perfect cubes is essential to grasp the fundamental structure of numbers.

To find perfect cubes, we cube natural numbers. Let's cube the first ten natural numbers to illustrate:
  • \(1^3 = 1\)
  • \(2^3 = 8\)
  • \(3^3 = 27\)
  • \(4^3 = 64\)
  • \(5^3 = 125\)
  • \(6^3 = 216\)
  • \(7^3 = 343\)
  • \(8^3 = 512\)
  • \(9^3 = 729\)
  • \(10^3 = 1000\)
You now have a clear list of perfect cubes beginning from 1. These numbers are particularly significant in certain arithmetic problems, including volume calculations, where length, width, and height are equal.
Integer Exponents
Exponents are a powerful tool in mathematics used to express repeated multiplication succinctly. An integer exponent is a whole number that indicates how many times a base number is to be multiplied by itself.

For example, with \(2^3\), the base is 2, and the exponent is 3, which tells us to multiply 2 by itself twice, resulting in 8. Here, exponents simplify the expression of large calculations.

There are some simple rules:
  • Any number to the power of 0 is 1, e.g., \(5^0 = 1\).
  • A number raised to the power of 1 is itself, e.g., \(8^1 = 8\).
  • Exponents can guide us in finding both perfect squares \((n^2)\) and perfect cubes \((n^3)\).
Understanding integer exponents is key when handling algebraic structures, scientific notation, and even understanding the scale of numerical data in everyday contexts.
Natural Numbers
Natural numbers are the fundamental numbers we use in counting and ordering. They start at 1 and go up infinitely. These numbers do not include zero or any negative numbers.

The sequence of natural numbers is one of the first sequences you learn in mathematics. Natural numbers are important because they serve as the foundation for more complex mathematical concepts.

Here's why natural numbers are handy:
  • They are used for counting objects, such as "There are 5 apples."
  • They include all numbers you need to find perfect squares and cubes, such as in exercises. For example, the perfect square of a natural number like 4 is \(4^2 = 16\).
  • Understanding and working with natural numbers prepares you for more advanced topics like algebra and calculus.
By mastering natural numbers, you build a base for understanding other types of numbers like integers, rational numbers, and beyond.

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Most popular questions from this chapter

A website selling nutritional supplements contains the following sentence: "Health is a function of proper nutrition." Explain what this statement means.

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(\frac{1}{4},\) passes through \((8,1)\)

Customer Service. \(\quad\) A software service hotline has found that on Mondays, the polynomial function \(C(t)=-0.0625 t^{4}+t^{3}-6 t^{2}+16 t\) approximates the number of callers to the hotline at any one time. Here, \(t\) represents the time, in hours, since the hotline opened at 8: 00 A.M. How many service technicians should be on duty on Mondays at noon if the company doesn't want any callers to the hotline waiting to be helped by a technician?

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope \(\frac{1}{5},\) passes through \((10,1)\)

Solve each equation. See Example \(10 .\) $$\frac{b}{5}=\frac{b-14}{2 b-16}-\frac{1}{2}$$
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