Problem 85 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 2 \cdot 5^{2}+4 \cdot 3^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 86.

Step 1: Evaluate the Exponents

First, we address the exponents in the expression. We have two exponents to evaluate: For \( 5^2 \), which is \( 5 \times 5 = 25 \).For \( 3^2 \), which is \( 3 \times 3 = 9 \).

Step 2: Multiply the Results

Next, we perform the multiplication for each part of the expression:For \( 2 \cdot 5^2 \), substitute the calculated value of \( 5^2 \) which is 25:\( 2 \times 25 = 50 \).For \( 4 \cdot 3^2 \), substitute the calculated value of \( 3^2 \) which is 9:\( 4 \times 9 = 36 \).

Step 3: Add the Products

Finally, we add the results from the previous step:\( 50 + 36 = 86 \).

Key Concepts

Evaluating Expressions

To evaluate an expression, you need to perform calculations methodically. This involves substituting values and performing operations as specified. In essence, you are simplifying the expression to find the final value it represents. When you evaluate expressions in algebra, it's crucial to understand the sequence in which you approach the expression. Breaking it down into individual components makes it easier to handle.

• First, identify the components of the expression such as numbers, variables, and operations.
• Next, apply the necessary mathematical operations following the rules of the order of operations.
• Finally, calculate the final result by completing all operations.

In the given exercise, we were tasked with evaluating the expression \(2 \cdot 5^{2} + 4 \cdot 3^{2}\). We evaluated by calculating each part and summing them together to find the total result.

Order of Operations

The order of operations is a set of rules that determines how to simplify or evaluate expressions involving multiple operations such as addition, subtraction, multiplication, division, and exponentiation. This order is crucial because depending on the sequence in which the operations are executed, the result can be entirely different. To help remember the order, many learners use the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

According to PEMDAS, computations should start with any calculations inside parentheses, followed by exponents, and so forth, moving from left to right. In our exercise, this rule helped ensure we tackled the exponents \(5^2\) and \(3^2\) before multiplication and addition. By following PEMDAS, we made sure that operations were performed in the correct sequence.

Exponents

Exponents represent repeated multiplication of a base number. When you see an expression like \(5^2\), it means that you multiply the base number 5 by itself one time, giving us \(5 \times 5\). Generally, \(a^n\) means the base \(a\) is multiplied by itself \(n\) times.
Exponential calculations have their own spot in the order of operations and need to be computed right after any parentheses and before performing multiplication and division.

• Exponents simplify calculations for large multiplication tasks.
• They help in expressing larger numbers compactly, as \(10^3\), which means \(10 \times 10 \times 10 = 1000\).
• In equations, solving for the exponents can often simplify the process of finding a solution.

In our original exercise, correctly determining the values of \(5^2\) and \(3^2\) as 25 and 9 respectively, was fundamental to accurately evaluating the expression.