Problem 40 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$5(-3)^{2}-2(-4)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(13\).

Step 1: Apply the Exponent

Begin by squaring the numbers inside the brackets, yielding \(5(-3)^{2}=5*9\) and \(-2(-4)^{2}=-2*16\). Thus, the expression now becomes \(5*9-2*16\).

Step 2: Simplify the Multiplication

Next, handle the multiplication in the expression. This operation results in the expression: \(45 - 32\).

Step 3: Perform the Subtraction

Finally, perform the subtraction. This results in the expression: \(13\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition or multiplication. For our exercise, we are working with a specific type of expression that involves both numbers and exponentiation. Understanding how to read and manipulate these expressions is crucial. Remember, each part of the expression has its own role, such as coefficients (numbers like 5), bases and exponents (like \((-3)^2\)), and terms that are added or subtracted. Always pay close attention to the placement of parentheses, as they indicate which parts of the expressions should be considered first according to the Order of Operations. In our example, the expression \(5(-3)^{2} - 2(-4)^{2}\) contains multiplication, subtraction, and exponentiation.

Exponentiation

Exponentiation refers to raising a number to a certain power, denoted as \(x^n\), where \(x\) is the base and \(n\) is the exponent. This operation means multiplying the base by itself as many times as the exponent indicates. In this context, understanding how to square a number, which is the same as raising it to the power of 2, is essential.
In the exercise, we have two bases, \(-3\) and \(-4\). Each is raised to the power of 2:

• \((-3)^{2} = (-3) \times (-3) = 9\)
• \((-4)^{2} = (-4) \times (-4) = 16\)

This process is crucial as getting the correct value after exponentiation ensures the accuracy of the remaining calculations.

Simplification Steps

Simplification involves breaking down the expression into simpler terms and performing operations systematically. Following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is key.
**Step-by-Step Guidance**
1. **Apply Exponents First**: As demonstrated, deal with the powers in \(5(-3)^{2}\) and \(-2(-4)^{2}\), giving \(45\) and \(-32\) respectively after simplification of multiplication.2. **Conduct Multiplication**: After managing the exponents, perform multiplication next, noting coefficients 5 and -2 in our problem.3. **Final Subtraction**: Lastly, subtract the resulting values - \(45 - 32\) to get \(13\).
Each of these steps must be approached in order, without skipping any, to ensure correctness in arriving at the final answer.