Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The simplified expression is 300.

Step 1: Simplify the Parentheses

Simplify the expression inside the parentheses first: \((4 \cdot 5)^{2}\) becomes \((20)^{2}\) and \(4 \cdot 5^{2}\) becomes \(4 \cdot 25\). The expression now looks like this: \(400-100\).

Step 2: Exponentiation

Next, deal with the exponent: \((20)^{2}\) simplifies to 400. The expression is now: \(400-100\).

Step 3: Final Calculation

Now, the subtraction can be performed, and the final simplified result is 300.

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number to the power of another number. In simpler terms, it's the repeated multiplication of a number by itself. In our exercise, we have an expression with exponentiation:

• (20)\(^{2}\) means multiplying 20 by itself.

Calculating the exponentiation of 20 and squaring it, we get:

• \(20 \times 20 = 400\).

It's essential to perform exponentiation before other operations like addition or subtraction, according to the order of operations.
This ensures that calculations are done correctly, leading to an accurate final result.

Simplification

Simplification is all about making an expression easier to understand and evaluate by performing the operations in the correct sequence. In this exercise, we are simplifying the expression:

• \((4 \cdot 5)^{2} - 4 \cdot 5^{2}\).

The first step is to simplify within the parentheses and perform exponentiation. Simplifying helps us get rid of complex expressions by breaking them down as follows:

• \( (20)^{2} - 4 \cdot 25\).

Next, we execute the calculations

• \(400 - 100\).

The result after these simplifications gives us the more straightforward value of 300.
This simplification process reduces the risk of error and helps in understanding the expression’s essence.

Parentheses

Parentheses indicate that the operations enclosed within them should be performed before any other operations outside. They play a crucial role in guiding the order of operations, ensuring that calculations are done in the intended sequence. In our exercise, parentheses are used as follows:

• \((4 \cdot 5)^{2}\).

The expression inside the parentheses, \(4 \cdot 5\), must be calculated first. Thus, \(4 \times 5\) equals 20. This result is then squared because it’s beneath the exponent.
Another part of the expression involves:

• \(4 \cdot 5^{2}\).

These parentheses ensure that multiplication within them is performed first before any external subtraction. This step-by-step calculation guarantees that no steps are skipped or interchanged, maintaining true mathematical order.

Expression Evaluation

Expression evaluation refers to the process of calculating the value of a mathematical expression following the established order of operations often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, the expression to be evaluated is:

• \((4 \cdot 5)^{2} - 4 \cdot 5^{2}\)

We begin by evaluating the expression inside the parentheses first, then moving to exponentiation. The detailed steps are as follows:

• Calculate the multiplication inside parentheses, simplifying \(4 \cdot 5\) to 20, leading to \(20^{2}\).
• Next, handle the exponentiation part by calculating \(20^{2} = 400\).
• Solve the additional multiplication \(4 \times 25 = 100\).
• Conclude by performing the final subtraction between 400 and 100.

Each step must comply strictly with the order of operations to avoid any misunderstanding and to assure the correctness of the final result.