Problem 48 - Elementary Algebra

Question

List all the cubes of the rational numbers in the set \\{-37,-17,0,17,37\\} .

Step-by-Step Solution

Answer

Cubes are \(-50653, -4913, 0, 4913, 50653\).

Step 1: Identifying the Task

We need to find the cube of each rational number in the set \(-37, -17, 0, 17, 37\). A cube of a number is that number multiplied by itself three times.

Step 2: Cube of -37

To find the cube of \(-37\), calculate \(-37 imes -37 imes -37\). First, compute \(-37 imes -37 = 1369\).Next, multiply the result by \(-37\): \(1369 imes -37 = -50653\). Thus, the cube of \(-37\) is \(-50653\).

Step 3: Cube of -17

Next, find the cube of \(-17\) by calculating \(-17 imes -17 imes -17\). First, compute \(-17 imes -17 = 289\).Then, multiply this result by \(-17\): \(289 imes -17 = -4913\). Therefore, the cube of \(-17\) is \(-4913\).

Step 4: Cube of 0

The cube of \(0\) is simple to calculate, as \(0 imes 0 imes 0 = 0\). Hence, the cube of \(0\) is \(0\).

Step 5: Cube of 17

To find the cube of \(17\), calculate \(17 imes 17 imes 17\). First, compute \(17 imes 17 = 289\).Next, multiply the result by \(17\): \(289 imes 17 = 4913\). Thus, the cube of \(17\) is \(4913\).

Step 6: Cube of 37

Finally, calculate the cube of \(37\) by evaluating \(37 imes 37 imes 37\). First, compute \(37 imes 37 = 1369\).Then, multiply this result by \(37\): \(1369 imes 37 = 50653\). Hence, the cube of \(37\) is \(50653\).

Step 7: Listing All Cubes

After calculating each, the cubes of the numbers in the set \(-37, -17, 0, 17, 37\) are \(-50653, -4913, 0, 4913, 50653\), respectively.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. Basically, they are numbers that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). Examples include numbers like \( \frac{1}{2} \), \( \frac{-3}{4} \), and even whole numbers like \( 5 \) since \( 5 \) can be expressed as \( \frac{5}{1} \).
Understanding rational numbers is crucial for various mathematical operations and number properties. In the exercise mentioning \(-37, -17, 0, 17, 37\), each of these numbers is a rational number because they can be represented as fractions with a denominator of 1.

• Math becomes more manageable when you realize every integer is also a rational number.
• Rational numbers include negative numbers, positive numbers, and zero.

When dealing with cubes of rational numbers, such as our example, it helps to understand that cubing a number means multiplying it by itself twice more (e.g., \(-37\) becomes \(-37 \times -37 \times -37\)).

Mathematical Operations

Mathematical operations are fundamental actions, such as addition, subtraction, multiplication, and division, used to solve problems and make calculations.
In the context of our exercise, we're focusing on the mathematical operation called "cubing" or "taking the cube" of a number. Let's break this down:

• **Cubing a Number**: This involves multiplying the number by itself twice (e.g., \( a^3 = a \times a \times a \)).
• **Negative Cubes**: When cubing a negative number, the result will also be negative because multiplying a negative by a negative results in a positive, but multiplying by another negative makes it negative again.

Cubing is often used in geometry to find the volume of cubes and other three-dimensional objects.
By practicing these operations, you become adept at finding powers of numbers and solving various mathematical problems. In our task, we've cubed numbers like \( -37 \) and \( 37 \) to obtain their respective cubes: \(-50653\) and \(50653\).
Mastering mathematical operations is key to advancing in math and other related fields.

Number Properties

Number properties are rules that numbers follow when we perform operations with them. These include the identity, inverse, and distributive properties, among others.
Focusing on properties relevant to our exercise:

• **Identity Property**: This states that any number multiplied by 1 remains unchanged (e.g., \( a \times 1 = a \)). This property is subtly at play when considering rational numbers as fractions.
• **Multiplicative Inverse Property**: For any rational number \( a \), there is another \( \frac{1}{a} \), such that \( a \times \frac{1}{a} = 1 \), assuming \( a eq 0 \).
• **Zero Property of Multiplication**: Any number multiplied by zero becomes zero. This helps explain why \( 0^3 = 0 \) in our example.

Understanding these properties can simplify working with numbers and help in performing complex mathematical operations. They allow us to predict the behavior of numbers in various scenarios, such as what happens when a number is cubed or multiplied by zero.