Problem 11 - Fundamentals of Mathematics

Question

Find each value. \(-\left(-|-4|^{2}\right)\)

Step-by-Step Solution

Answer

The value is 16.

Step 1: Calculate Absolute Value

First, find the absolute value of -4. The absolute value of a number is its distance from zero on the number line, regardless of direction. Thus, |-4| = 4.

Step 2: Square the Absolute Value

Now, square the absolute value found in Step 1. Calculate 4^2, which is 16.

Step 3: Negate the Squared Value

Negate the squared value obtained in Step 2 to solve the operation inside the negative parenthesis. Thus, the expression becomes - 16.

Step 4: Simplify Expression

Finally, negate the result from Step 3 to simplify and obtain the final value. Hence, -(-16) = 16.

Key Concepts

Negation

Negation is a fundamental concept in mathematics, often represented by the negative sign (-). When you negate a number, you are essentially finding its additive inverse. For example, the negation of 5 is -5, and vice versa, because when you add them together, you get zero:

• 5 + (-5) = 0

Negation can be thought of as flipping the position of a number on the number line from one side of zero to the other. If the number is positive, its negation will be negative, and if it is negative, its negation will be positive.
When using negation in expressions, be attentive to signs. For example, negating -16, written as -(-16), results in a positive 16. This is because (-)(-) = +. It might help to remember this rule:

• Negating a negative gives a positive.
• Negating a positive gives a negative.

Squaring Numbers

Squaring a number means multiplying it by itself. For instance, when you square the number 4, you get \(4 \times 4 = 16\). Squaring is a powerful operation because it always results in a positive value, as multiplying two positive numbers results in a positive.
Mathematically, any positive number squared is positive, and surprisingly, any negative number squared is also positive. This happens because multiplying two negative values results in a positive. For example:

• \((-3)^2 = (-3) \times (-3) = 9\)

Squaring is a common operation in many areas of math and science, often used to find areas, variances, and more. It's crucial to remember:

• Squaring always yields a non-negative result.
• It reflects a number's symmetry is centered around zero on the number line.

Number Line

A number line is a basic yet powerful visual tool used in mathematics to represent numbers. It allows for easy visualization of concepts such as distance, addition, subtraction, and absolute values.
A typical number line features a horizontal line with zero at the center. Positive numbers extend to the right, while negative numbers stretch to the left. Key uses include:

• Locating numbers: Helps in visualizing where numbers fall.
• Describing order and value: Numbers to the right are greater than those to the left.
• Understanding distance: For absolute value, the distance from zero is critical regardless of direction.

For example, to find the absolute value of -4 on the number line, you consider its distance from zero, which is 4. This insight helps in understanding the problem of finding \(-|-4|\), where the absolute value simplifies to a positive number before performing any arithmetic operations.