Problem 38 - Fundamentals of Mathematics

Question

Determine each of the values. $$ -[|-10|-6]^{2} $$

Step-by-Step Solution

Answer

The final result is -16.

Step 1: Resolve Absolute Values Inside Brackets

First, evaluate the innermost expression using absolute values. The expression is \(|-10|\), which evaluates the absolute value of -10. Remember, absolute value is the distance of a number from zero on the number line, so \[-10\] -> \[10\]. Thus, \([|-10|] = 10\).

Step 2: Apply Additional Operations Inside Brackets

Next, subtract 6 from the result obtained in Step 1. The expression becomes \(10 - 6\).So, \[10 - 6 = 4\].

Step 3: Square the Result

Square the result obtained in Step 2. The expression to evaluate now is \[4^2\].Calculate the square of 4: \[4 \times 4 = 16\].

Step 4: Apply Negative Sign

Now, apply the negative sign in front of the entire expression. The expression will become \(-[...]\), which we have as -16.

Key Concepts

Evaluate Expressions

Evaluating expressions involves simplifying an expression by performing all necessary operations to reduce it to a single numerical value. In this exercise, the task is to find the value of the expression \(-[|-10|-6]^2\). First, understanding absolute values is crucial. An absolute value represents the distance of a number from zero on the number line and is always non-negative. Therefore, \(|-10|\) is the same as 10. Once the absolute value is calculated, further operations like subtraction, squaring, and adjusting for any negative signs are carried out to reach the final result. Ensure each step is accurate and keep a close eye on the order of operations. This way, you'll find the correct value of a mathematical expression.

Step-by-Step Solutions

Breaking down a mathematical problem into smaller, more manageable steps makes finding a solution easier. Let's look at the complete step-by-step process used in this exercise:

• Step 1: Begin by resolving any absolute values. Here, \(|-10|\) becomes 10 because the absolute value makes any negative number positive.
• Step 2: Perform subtraction within the brackets next. Subtract 6 from 10 to get 4.
• Step 3: Next, square the result from the subtraction. Squaring 4 gives \(4 imes 4 = 16\).
• Step 4: Finally, apply the negative sign to the entire expression, resulting in -16.

Following these steps ensures you consider each mathematical operation systematically. This methodical approach avoids mistakes and confusion.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks of evaluating expressions. In this context, understanding the correct order of operations is essential: often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
Within this exercise, parentheses take precedence, where resolving inside brackets is necessary before tackling any further arithmetic. Understanding exponents is crucial where squaring a number, such as 4 in \(4^2\), means multiplying the number by itself.
Lastly, the negative sign at the front of the expression dictates the need to reverse the sign of the entire result. Practicing this orderly approach enhances your ability to handle much more complex expressions with ease.