Problem 47 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$(1.5)^{2}+(2.5)^{2}+(3.5)^{2}$$

Step-by-Step Solution

Answer

The result is 20.75.

Step 1: Calculate Squares

First, calculate each squared term separately.\[(1.5)^2 = 1.5 \times 1.5 = 2.25\]\[(2.5)^2 = 2.5 \times 2.5 = 6.25\]\[(3.5)^2 = 3.5 \times 3.5 = 12.25\]

Step 2: Sum the Squares

Now, sum the squared values obtained in the first step.\[2.25 + 6.25 + 12.25\]

Step 3: Add Step-by-Step

Start by adding the first two squared results:\[2.25 + 6.25 = 8.5\]Then add the last squared result:\[8.5 + 12.25 = 20.75\]

Key Concepts

Understanding Exponents

Exponents are a mathematical way to express repeated multiplication. They are used when a number is multiplied by itself a certain number of times. The small raised number or "exponent" indicates how many times the base number is being multiplied by itself. For example, in \((1.5)^2\), the number 1.5 is the base, and 2 is the exponent. This expression means 1.5 is multiplied by itself once to obtain 2.25. Similarly:

• \((2.5)^2 = 2.5 \times 2.5 = 6.25\)
• \((3.5)^2 = 3.5 \times 3.5 = 12.25\)

Understanding exponents is crucial because they form a fundamental part of many mathematical problems, letting us simplify expressions and solve equations more efficiently.

Mastering Addition

Addition is one of the fundamental arithmetic operations. It involves combining two or more numbers to get their total or sum. In the given exercise, once the exponents were calculated, you follow straightforward addition to find the sum of these resultant values. The key here is to align numbers by their decimal points to ensure accuracy.After calculating the squares:

• First, add \(2.25 + 6.25 = 8.5\)
• Next, add the result, 8.5, with 12.25: \(8.5 + 12.25 = 20.75\)

Practicing addition helps develop a strong foundation in mathematics, pivotal for solving more complex mathematical problems.

Grasping Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. When handling problems like the one in this exercise, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells you the correct sequence to solve the problem:

• First, handle operations inside parentheses or brackets.
• Next, calculate any exponents.
• Then, perform multiplication and division from left to right.
• Finally, perform addition and subtraction from left to right.

This exercise uses these operations by calculating exponents first and then using addition, showing a practical example of following the order: calculate squares before summing them, resulting in the total 20.75. Understanding and mastering these operations is vital for tackling various mathematical challenges.