Problem 35 - Fundamentals of Mathematics

Question

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

Step-by-Step Solution

Answer

The value is 100.

Step 1: Evaluate Absolute Values

First, calculate the absolute values of both numbers inside the expression. The absolute value of a number is its distance from zero on the number line, regardless of direction.\(|-6| = 6\)\(|4| = 4\)

Step 2: Add the Absolute Values

Add the results of the absolute values obtained in Step 1:\(6 + 4 = 10\)

Step 3: Square the Sum

Now, square the result from Step 2 to find the final answer:\((10)^2 = 100\)

Key Concepts

Square Numbers

In mathematics, square numbers are fundamental concepts that involve multiplying a number by itself. They represent quadratic growth and have various applications in different fields including physics and engineering. When we talk about squaring a number like 10, we express it as the number raised to the power of 2, which is written in mathematical notation as \(10^2\). This operation is important because it helps in understanding growth patterns and can simplify complex equations.
Squaring numbers is a common operation to solve many algebraic equations and problems. Here's a straightforward step to calculate a square number:

• Identify the number you need to square.
• Multiply the number by itself.
• Record the result as the square of the original number.

For example, in our exercise, after adding the values, we had to square 10 which results in \(10 \times 10 = 100\). Understanding square numbers is crucial in mastering more complex mathematical concepts.

Addition

Addition is one of the basic operations in mathematics, involving the combination of two or more numbers to get a cumulative total. It lays the foundation for understanding more advanced operations like multiplication and exponentiation.
Understanding addition is crucial, and here's how it works:

• It is the process of calculating the total of two or more numbers.
• The sum is usually greater than the original numbers unless adding zero.
• An essential property of addition is its commutativity, meaning \(a + b = b + a\), where \(a\) and \(b\) are numbers.

In the exercise, we used addition after calculating the absolute values of \(6\) and \(4\). Adding these absolute values gave us \(10\), which is necessary to proceed with finding the square of this sum. This basic operation is fundamental in everyday calculations and higher-level mathematics alike.

Step-by-Step Solution

Approaching problems with a step-by-step solution is a powerful method to tackle complex exercises. It helps in breaking down the problem into manageable parts, ensuring understanding and accuracy.
Let's see how the step-by-step method applies to our exercise:

• Step 1: Calculate the absolute values of the numbers provided. This involves recognizing that absolute value represents a number's distance from zero without considering direction.
• Step 2: Add the absolute values found in the previous step. This operation combines them to simplify any subsequent calculations.
• Step 3: Square the sum obtained from addition to arrive at the final solution.

By following a step-by-step approach, you ensure that each part of the problem is addressed methodically. This logical progression not only aids in understanding the process but also in discovering similar patterns across different mathematical problems.