Problem 18 - Beginning and Intermediate Algebra

Question

Evaluate. \((0.07)^{2}\)

Step-by-Step Solution

Answer

0.0049

Step 1: Understand the Question

We are asked to evaluate \((0.07)^{2}\), which means we need to find the value of \(0.07\) multiplied by itself.

Step 2: Write the Multiplication

Express \((0.07)^{2}\) as \(0.07 \times 0.07\). This means multiplying 0.07 by 0.07.

Step 3: Multiply the Numbers

Conduct the multiplication: \(0.07 \times 0.07 = 0.0049\). Multiply by aligning the figures: 7 times 7 is 49, and since each number is two places after the decimal, the result must be four places after the decimal.

Key Concepts

Decimal Multiplication

Decimal multiplication involves multiplying numbers that have digits after the decimal point. Unlike whole numbers, decimals have fractional parts represented by digits to the right of the decimal point, which require careful attention when performing arithmetic operations.
Here are some key points to consider when multiplying decimals:

• Ignore the decimal point and multiply the numbers as if they were whole numbers.
• Count the total number of digits to the right of the decimal points in both numbers being multiplied.
• Place the decimal point in the product so that it has the same number of decimal places as the sum of the decimal places in the factors.

For example, when multiplying 0.07 by 0.07, first perform the multiplication without considering the decimals: 7 times 7 equals 49. Since there are four digits after the decimals in total, place the decimal in the product to yield 0.0049.
Being precise and systematic helps in avoiding common mistakes, such as misplacing the decimal point.

Square of a Decimal

Finding the square of a decimal means multiplying the decimal number by itself. This concept applies the same principles as the square of whole numbers but requires extra care with decimal placement.
When squaring a decimal:

• Write the decimal as a multiplication of the number by itself (e.g., \((0.07)^{2} = 0.07 \times 0.07\)).
• Perform the multiplication while ignoring decimal points first, similar to whole numbers.
• Count all decimal places in the initial decimal, and ensure the squared result matches this count in placement of the decimal in the answer.

Following these steps, \(0.07 \times 0.07 = 0.0049\). The result should be smaller than the original decimal since we are multiplying a fraction by itself, leading to an even smaller fraction.

Basic Arithmetic Operations

Basic arithmetic operations are foundational mathematical procedures, including addition, subtraction, multiplication, and division. These operations are essential for solving everyday problems and serve as building blocks for more advanced mathematical concepts.
Understanding basic operations starts with knowing how each operation affects numbers:

• Addition: Combine numbers to find their total.
• Subtraction: Determine the difference between numbers by removing one from another.
• Multiplication: Find the product by combining equal groups of a number.
• Division: Split a number into equal parts.

While performing these operations with decimal numbers, it is crucial to align decimal points properly. For multiplication and division especially, the placement of the decimal point in the result must reflect the correct operation outcome based on the number of decimal places involved in the calculation. This precision ensures accurate mathematical representations.