Problem 43 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$(-5)^{2}+20 \div 4$$

Step-by-Step Solution

Answer

The simplified expression is 30.

Step 1: Evaluate Exponents

First, address any exponents in the expression. In this case, evaluate \[(-5)^2 = (-5) imes (-5) = 25.\] Now the expression becomes \[25 + 20 \div 4.\]

Step 2: Division

Next, handle the division operation. Divide 20 by 4:\[20 \div 4 = 5.\] This simplifies the expression further to \[25 + 5.\]

Step 3: Addition

Finally, perform the addition:\[25 + 5 = 30.\] This is the simplified expression.

Key Concepts

Exponents

Exponents are a fundamental concept in algebra, representing repeated multiplication of a number by itself. When you see an expression like \((-5)^2\), it means that -5 is multiplied by itself. So, the calculation is \((-5) \times (-5)\), which equals 25. It's important to remember that squaring a negative number results in a positive product. This happens because multiplying two negative numbers results in a positive.
When dealing with exponents combined with other operations, use the order of operations rule, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This rule tells us to handle the exponents before moving on to other operations.
Understanding how to correctly evaluate expressions with exponents will help simplify complex mathematical problems. Remember, always break down the operation in steps, ensuring that the exponent is addressed first before proceeding with division or addition.

Division

Division is the process of splitting a number into equal parts. In the expression given, after evaluating the exponents, we tackle the division part. The expression at this stage is \(25 + 20 \div 4\).
According to the order of operations (PEMDAS), division comes before addition. So, you need to divide 20 by 4, which gives 5. This changes the expression to \(25 + 5\).
A crucial point to remember is how division is performed before addition when simplifying expressions. Skipping the proper order might lead to incorrect results. By practicing, students can become adept at recognizing and applying the correct steps, ensuring they arrive at beautiful simplified expressions.

Addition

Addition is one of the simplest and fundamental arithmetic operations, where two numbers are combined to make a larger one.
After handling exponents and division, you're left with the final step in our example: \(25 + 5\).
Performing this operation, 25 plus 5 equals 30. Addition here combines previous calculations to create a final result.

• Ensure you follow the order of operations.
• Always double-check your work to avoid simple mistakes.

Understanding the role of addition as the final step confirms the importance of proceeding sequentially. With practice, students will become comfortable with identifying and executing the steps in the right order.