Problem 57 - Prealgebra

Question

Perform the indicated operations. $$12 \times 12$$

Step-by-Step Solution

Answer

The product of 12 and 12 is 144.

Step 1: Understanding the Operation

The operation given is multiplication, which involves finding the total when one number is taken a certain number of times. We need to multiply 12 by 12.

Step 2: Apply Multiplication Formula

To multiply two numbers, align them in a column format and multiply each digit step-by-step from right to left. However, since the numbers are the same, we can use the formula of a square: \[ a^2 = a \times a \] for \( a = 12 \).

Step 3: Use Square Formula

Calculate the square of 12. Use: \[ 12^2 = 12 \times 12 \]. This can be calculated directly or by using known multiplication tables.

Step 4: Calculate the Result

Using direct multiplication or a multiplication table, when 12 is multiplied by 12, the result is 144.

Key Concepts

Multiplication Tables

Multiplication tables are essential tools in the world of arithmetic, helping students quickly find the product of two numbers. They are essentially grids that show the results of multiplying different numbers together. For example, the number 12's row in the multiplication table shows the results of multiplying 12 by each other number, like 12, 24, 36, and so on.

Using multiplication tables is handy because it:

• Saves time by providing quick answers to multiplication problems.
• Reinforces memory through repetitive learning.
• Builds a foundation for more complex mathematical tasks.

Although calculators and computers can quickly give you answers, understanding and memorizing multiplication tables help in developing mental math skills. Next time you see a multiplication problem, try using a table to find the answer swiftly!

Square Numbers

Square numbers are the result of multiplying a number by itself. It is essentially a form of repeated multiplication. In the exercise, calculating the square of 12, denoted as \(12^2\), is essentially finding \(12 \times 12\).

Square numbers have interesting properties:

• Each square number is non-negative.
• The difference between consecutive square numbers increases as the numbers increase.
• They are visually represented as perfect squares when laid out in grids or arrays.

For example, the square of 12 is 144. So, when you come across a question asking for a number's square, remember it's just that number multiplied by itself.

Arithmetic Operations

Arithmetic operations form the foundation of all mathematics. The basic operations include addition, subtraction, multiplication, and division. Each of these operations serves a unique purpose in calculations. Our primary focus here is multiplication, an operation that determines how many times a number is added to itself.

Some key points about multiplication:

• It is one of the four basic arithmetic operations.
• It can be thought of as repeated addition.
• It is commutative; you can change the order of the numbers and still get the same result. For example, \(12 \times 12 = 12 \times 12\).
• The result of multiplying two positive numbers is always positive.

Understanding the concept of multiplication can significantly ease other more advanced mathematical operations, making arithmetic operations a crucial part of learning mathematics.