Problem 48 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$(1.1)^{2}+(2.1)^{2}+(3.1)^{2}$$

Step-by-Step Solution

Answer

The result is 15.23.

Step 1: Square each term

Calculate the square of each term separately: \((1.1)^2 = 1.21\)\((2.1)^2 = 4.41\)\((3.1)^2 = 9.61\)

Step 2: Add the squared terms

Add the squared values together to find the total: \(1.21 + 4.41 + 9.61\)

Step 3: Perform the addition

First add \(1.21 + 4.41 = 5.62\).Then add \(5.62 + 9.61 = 15.23\).

Key Concepts

Squaring Numbers

Squaring a number, simply put, means multiplying the number by itself. This operation is fundamental when dealing with exponents and can be written in the form \(a^2\), where \(a\) is the number being squared.

• Squaring even a small decimal like \(1.1\) or \(2.1\) involves multiplying \(1.1 \times 1.1\) and \(2.1 \times 2.1\) respectively.
• For example, \((1.1)^2 = 1.1 \times 1.1 = 1.21\).
• Similarly, \((3.1)^2 = 3.1 \times 3.1 = 9.61\).

When squaring a number, each part of the number affects the answer, and results may include decimals as well, which makes dealing with decimals in squaring slightly more complex, but still straightforward once you grasp the multiplication part.
This principle also applies to negative numbers, which yield positive results because a negative times a negative equals a positive. Thus, \((-a)^2 = a^2\).

Addition of Decimals

Adding decimals can seem tricky at first, but it's just as simple as regular addition once your numbers are properly aligned. The key is to line up the decimal points before adding, then proceed just as you would with whole numbers.

• Take each pair of numbers, such as \(1.21\) and \(4.41\), line them up vertically by the decimal point, and add column by column.
• Start from the rightmost column, moving to the left, carrying over any value greater than 10 as needed, just like in traditional addition.
• After summing \(1.21 + 4.41 = 5.62\), continue by adding \(5.62 + 9.61\) in the same manner, yielding a total of \(15.23\).

Keep in mind, precision in the placement of decimal points is crucial to obtaining the correct sum. Mistakes often occur when numbers are improperly aligned. The outcome is always about ensuring that each decimal is added to the correct positional place value.

Mathematical Operations

Mathematical operations such as squaring and addition follow specific rules, known as the "order of operations." Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), these rules determine the sequence in which operations should be performed in complex expressions.

• First, handle any exponents. For the given problem, this meant calculating \((1.1)^2, (2.1)^2,\) and \((3.1)^2\).
• Next, perform addition of these squared results: first \(1.21\) added to \(4.41\), and then that sum added to \(9.61\).
• Following the order of operations ensures that the calculation proceeds logically and accurately, aligning with mathematical conventions.

Understanding these rules aids in solving more complex problems systematically, reducing the chance of errors.
Practicing this order on simpler numbers helps build familiarity, making more complex calculations easier to handle.