Problem 27 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ -5^{2}-20 \div 4-2 $$

Step-by-Step Solution

Answer

The expression evaluates to -32.

Step 1: Apply Exponentiation

The expression begins with evaluating the part related to the exponent. Here, we need to square the number 5. Note the negative sign before the 5 is not part of the base.Evaluate: \[-5^2 = -(5^2) = -(25) = -25\]

Step 2: Perform Division

Next, handle the division operation in the expression. Divide 20 by 4.Evaluate: \[20 \div 4 = 5\]

Step 3: Evaluate Remaining Operations

Now, substitute the evaluated parts back into the expression and perform the remaining operations. Start with the leftmost operation, which is addition/subtraction.Expression becomes: \[-25 - 5 - 2\]First, \[-25 - 5 = -30\]And then, \[-30 - 2 = -32\]

Step 4: Final Result

The entire original expression evaluates to: \[-32\]

Key Concepts

Order of Operations

When we tackle algebraic expressions, it's crucial to follow the Order of Operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Applying this order ensures that expressions are simplified correctly. In our exercise, despite the absence of parentheses, we still adhere to evaluating exponentiation before division and subtraction.

• First, solve any calculations involving exponents.
• Next, proceed to division and multiplication as they appear from left to right.
• Finally, handle addition and subtraction from left to right.

Following these rules, we maintain consistency and accuracy in evaluating the expression. This prioritization helps clarify which operations to perform first, avoiding confusion and errors.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is raised to the power of the exponent, meaning the base is multiplied by itself the number of times indicated by the exponent. For example, in our expression \(-5^2\), only the 5 is squared.

• The exponent "2" means you multiply the base, 5, by itself: \(5 \times 5 = 25\).
• However, the negative sign before the expression must be considered separately as it is not inside parentheses. This means we interpret it as \(-(5^2)\).
• Thus, the expression evaluates to \(-25\), not (+25), which is a common mistake.

Paying close attention to the placement of negative signs and parentheses is important in getting the correct result.

Division and Subtraction

After handling exponents, division takes precedence over subtraction. In this context, division is the process by which one number is divided by another, effectively determining how many times the divisor can fit into the dividend. From the given exercise:

• We have \(20 \div 4 = 5\), which simplifies our expression further.

The subtraction operation is straightforward once division and exponentiation are resolved. It simply involves taking away one quantity from another, processed from left to right, while maintaining any negative signs in the correct position:

• First, solve \(-25 - 5 = -30.\)
• Then, continue with \(-30 - 2 = -32.\)

This sequential approach ensures that each step is clear and the final outcome is arrived at systematically, resulting in \(-32\). Remember, in subtraction especially, the order in which you perform calculations makes a difference in the result.