Problem 47 - Fundamentals of Mathematics

Question

Determine each value. $$ |-4|^{2} $$

Step-by-Step Solution

Answer

The value is 16.

Step 1: Understanding the Absolute Value

To solve the expression \(|-4|^2\), we first need to understand that absolute value represents the distance of a number from zero on the number line, regardless of its sign. Therefore, the absolute value of -4 is obtained by removing the negative sign, resulting in 4.

Step 2: Compute the Absolute Value

Determine the absolute value: \(|-4| = 4\). This simplifies our original expression to \(4^2\).

Step 3: Exponentiate the Absolute Value

Now, compute the square of the absolute value obtained in Step 2. This involves multiplying the number by itself: \(4^2 = 4 \times 4 = 16\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to a certain power, represented as an exponent. In our exercise, when we see the notation \(4^2\), it means we are raising 4 to the power of 2. This is equivalent to multiplying the number by itself. Exponentiation is often used to simplify calculations or express large numbers concisely.

• For any base \( a\) and exponent \( n\), the expression \( a^n\) indicates that \( a\) is multiplied by itself \( n\) times.
• In this case, \( 4^2 = 4 \times 4 = 16\), which involves simple multiplication.

Understanding exponentiation helps in solving expressions quickly, like in our case where we found \( 4^2 = 16\) effortlessly.

Number Line

The number line is a visual representation of numbers arranged sequentially along a straight line. It assists in understanding the concept of order and relative position of numbers, including negative, zero, and positive numbers. The number line continues endlessly in both the positive and negative directions.

• Zero is the central point on the number line.
• Numbers to the right of zero are positive, while numbers to the left are negative.
• Understanding number lines is crucial for visualizing the concept of absolute value.

In our exercise, by using the number line, we can see that the number -4 is four units away from zero to the left. This ties into the concept of absolute value.

Distance from Zero

Distance from zero refers to how far a number is located from zero on the number line, ignoring the direction. This concept is crucial when dealing with absolute values, as it represents only magnitude, not the sign.

• For any number \( a\), the distance from zero is always non-negative.
• The absolute value is the numerical value representative of this distance without regard to sign.
• In our example, the absolute value of \(-4\) is 4 because it is four spaces away from zero.

Grasping the idea of distance from zero helps clarify why absolute values ignore negative signs, turning \( |-4| \) to 4.

Multiplication

Multiplication is one of the four basic operations of arithmetic, which combines groups of equal sizes into a larger total. It is an essential operation for performing exponentiation. In exponentiation, we use multiplication to combine multiple instances of a number, driven by the power or exponent.

• For example, in \( 4^2 \), we multiply 4 by itself: \( 4 \times 4 \).
• This is different from addition as it accumulates the result much faster, effectively summarizing repeated additions.
• Understanding multiplication is key to solving expressions with exponents easily.

In the given problem, multiplication helped calculate the exponential value of the absolute number, resulting in \( 4^2 = 16 \).