Problem 46 - Elementary Algebra

Question

List all the cubes of the integers from -10 to 0 .

Step-by-Step Solution

Answer

The cubes are -1000, -729, -512, -343, -216, -125, -64, -27, -8, -1, and 0.

Step 1: Understand the Cube Function

The cube of a number is obtained by multiplying the number by itself twice. For a number \( n \), the cube is calculated as \( n^3 = n \times n \times n \). We will use this function to find the cubes of the integers in the given range from -10 to 0.

Step 2: List the Integers

Identify and list the integers from -10 to 0. These integers are: -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, and 0.

Step 3: Calculate Cubes for Each Integer

Compute the cube of each integer in the list from Step 2. - \((-10)^3 = -10 \times -10 \times -10 = -1000\)- \((-9)^3 = -9 \times -9 \times -9 = -729\)- \((-8)^3 = -8 \times -8 \times -8 = -512\)- \((-7)^3 = -7 \times -7 \times -7 = -343\)- \((-6)^3 = -6 \times -6 \times -6 = -216\)- \((-5)^3 = -5 \times -5 \times -5 = -125\)- \((-4)^3 = -4 \times -4 \times -4 = -64\)- \((-3)^3 = -3 \times -3 \times -3 = -27\)- \((-2)^3 = -2 \times -2 \times -2 = -8\)- \((-1)^3 = -1 \times -1 \times -1 = -1\)- \((0)^3 = 0 \times 0 \times 0 = 0\)

Step 4: List the Results

The cubes of the integers from -10 to 0 are: -1000, -729, -512, -343, -216, -125, -64, -27, -8, -1, and 0.

Key Concepts

Understanding the Cube Function

When we talk about the cube function, we are referring to the mathematical process where a number is multiplied by itself twice. It's as simple as choosing a number, let's call it \( n \), and then considering \( n \times n \times n \). This operation is represented in mathematics as \( n^3 \). The result we obtain is called the cube of that particular number.
This function is fundamental in math because it shows us how a number grows exponentially when extended into the third dimension.

• The operation is repeated multiplication, showcasing exponential growth.
• It's an important concept in geometry, as it helps find volumes of cubes.
• The function applies to all real numbers, both positive and negative.

Grasping this function can be made easier by practicing with small numbers, which helps visualize how quickly numbers increase when cubed.

Exploring Negative Integers

Negative integers are numbers less than zero. They occupy the left side of the number line and essentially represent a lack or deficit. Understanding how these numbers behave requires us to adjust our thinking slightly, as their arithmetic can yield results that might seem unusual at first.
When we apply the cube function to negative integers, we multiply these numbers as follows: a negative number cubed results in a negative value. The reason is simple: the multiplication of three negative numbers results in a negative product.

• Each multiplication of a negative flips the sign: two negatives make a positive, but three return it to negative.
• Calculating cubes with negative integers can often result in large negative numbers.

Negative cubes are an interesting part of mathematics as they showcase how the same operation can produce dramatically different results based on the input values.

Creating a Step-by-Step Solution

The step-by-step solution is a methodical approach to solving a problem by breaking it down into smaller, more manageable parts. In this context, we follow a sequence to calculate the cubes of integers from -10 to 0. This process ensures that we don’t miss any detail and can easily find our way back if a mistake is made.
Breaking down the solution:

• Identify the Range: First, list the range of integers. Understanding what numbers you are working with is crucial.
• Apply the Cube Function: For each integer, apply the cube function \( n^3 \) to calculate their cubes.
• Verify the Results: Confirm computations by considering the multiplication's outcome, especially for negative integers.

This structured approach simplifies complicated problems and builds a systematic mindset conducive to solving a variety of mathematical challenges.