Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$(4-6)^{2}-(5-9)^{2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( -12 \).

Step 1: Evaluate Expressions inside Parentheses

Calculate the result of the subtraction operations within the parenthetical expressions. So, \((4-6)\) becomes \(-2\), and \((5-9)\) becomes \(-4\). Now the expression will look like \((-2)^{2}-(-4)^{2}\).

Step 2: Apply the Exponentiation

Now, square the numbers. \((-2)^{2}\) becomes \(4\), and \((-4)^{2}\) becomes \(16\). Thus, the expression simplifies to \(4 - 16\).

Step 3: Perform Subtraction

Lastly, perform the subtraction operation. Subtracting 16 from 4 results in \(-12\).

Key Concepts

Evaluating Expressions

When working with mathematical expressions, it's crucial to follow the order of operations to ensure accurate results. The order of operations is typically remembered through the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Evaluating expressions involves simplifying these operations one step at a time. Begin with the innermost parentheses and work your way outward, respecting the hierarchy of operations. In the given problem, parentheses are used to indicate which operations to perform first. This means you should focus on solving everything inside the parentheses before moving on.

In our example,

• Start by evaluating \(4 - 6\) and \(5 - 9\)
• These simplify to \(-2\) and \(-4\) respectively

Completing this step turns the original problem into the simplified expression \((-2)^{2} - (-4)^{2}\).

By clearly defining each step, you eliminate confusion and increase the accuracy of your calculations.

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base and the exponent. The exponent denotes how many times the base will be multiplied by itself. For example, in \(a^2\), "a" is the base, and the "2" is the exponent, indicating that "a" should be squared (multiplied by itself once).

In our exercise, the exponentiation occurs as follows:

• \((-2)^{2}\) is squared to produce the answer \(4\). This means \(-2 imes -2 = 4\)
• \((-4)^{2}\) is squared to yield \(16\) because \(-4 imes -4 = 16\)

Squaring negative numbers can initially be tricky. Remember that any number squared (whether negative or positive) results in a positive number because multiplying two negative numbers results in a positive product. That's why both results here are positive.

Understand that correct exponentiation is key to simplifying expressions accurately and is essential when handling more complex algebraic problems.

Subtraction

Subtraction is one of the basic arithmetic operations, where you take the difference between two numbers. It's the process of "taking away" or "reducing."

In this particular problem, subtraction is the final step after exponentiation. You are tasked with subtracting the result of one squared term from another:

• The expression simplifies to \(4 - 16\).
• This represents taking the value 16 away from 4.

To perform this, align the numbers and carry out the subtraction as you would under normal arithmetic rules. Here, it equates to a negative result: \(-12\).

Visualizing subtraction as a movement on the number line can help. If you start at 4 and move left past zero to subtract 16, you ultimately land on -12. This clarifies the process and signifies the importance of following operations precisely to obtain the correct result.