Problem 37 - Fundamentals of Mathematics

Question

Determine each of the values. $$ (|4|+|-6|)^{2}-(|-2|)^{3} $$

Step-by-Step Solution

Answer

The value is 92.

Step 1: Calculate Absolute Values

Begin by evaluating the absolute values in the expression. The absolute value of a number is its distance from zero, ignoring the sign. - \(|4| = 4\) - \(|-6| = 6\) - \(|-2| = 2\)

Step 2: Sum of Absolute Values

Add the absolute values of \(4\) and \(-6\).\(|4| + |-6| = 4 + 6 = 10\)

Step 3: Square the Sum

Square the result from Step 2.\((|4| + |-6|)^2 = 10^2 = 100\)

Step 4: Cube the Absolute Value of -2

Calculate the cube of the absolute value of \(-2\).\((|-2|)^3 = 2^3 = 8\)

Step 5: Subtract the Cubed Value from the Squared Value

Subtract the result from Step 4 from the result from Step 3.\((|4| + |-6|)^2 - (|-2|)^3 = 100 - 8 = 92\)

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics, and it's all about understanding the distance of a number from zero on the number line, regardless of direction. Absolute value turns any negative number into a positive number. Consider these points when working with absolute values:

• Symbol: The absolute value of a number \( x \) is written as \(|x|\).
• Basic Rule: \(|x|\) is the same as \(x\) if \(x \geq 0\), and it is \(-x\) if \(x < 0\).
• Example: For the number \(-6\), the absolute value \(|-6|\) is \(6\) because it's the distance from zero.

Understanding this concept is crucial as it forms the first step in solving expressions like the one given.

Mastering Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations are carried out. This is essential for consistent results in mathematics. Remember the acronym PEMDAS:

• **P**arentheses
• **E**xponents (including powers and roots)
• **MD** (Multiplication and Division, from left to right)
• **AS** (Addition and Subtraction, from left to right)

In the exercise, after computing the absolute values, the operations follow with exponentiation (squaring and cubing) before subtraction. Ensuring the correct order is followed prevents mistakes and the wrong answer.

Squaring Numbers

Squaring a number involves multiplying the number by itself. Here are some important points about squaring:

• Squaring a number \( x \) is written as \(x^2\).
• The result of a square is always positive since a negative number squared turns positive. For example, \((-5)^2 = 25\).
• In our exercise, \((|4| + |-6|)^2 = 10^2 = 100\).

Understanding how squaring works help simplify expressions and equations in various mathematical problems effectively.

Exploring Cubing Numbers

Cubing a number means raising it to the third power, which involves multiplying the number by itself three times. Here’s how to think about cubing:

• Cubing a number \( x \) is denoted as \(x^3\).
• A cube retains the sign of the original number—negative numbers remain negative and positive numbers remain positive. For example, \((-3)^3 = -27\).
• In the given example, \((|-2|)^3 = 2^3 = 8\).

Cubing is an exponential operation that is crucial in various contexts, including volume calculations and transformations in geometry.