Problem 66 - Prealgebra

Question

Find the value of each expression. $$21^{2}$$

Step-by-Step Solution

Answer

The value of the expression \(21^2\) is 441.

Step 1: Understand the Task

The task is to find the square of 21, which means multiplying 21 by itself.

Step 2: Break Down the Multiplication

To find the square of 21, we need to calculate the product of 21 with itself: \(21 \times 21\).

Step 3: Calculate the Multiplication

Perform the multiplication step-by-step: first multiply \(21\) by \(20\) (\(21 \times 20 = 420\)) and then by \(1\) (\(21 \times 1 = 21\)), and then add the results together: \(420 + 21 = 441\).

Step 4: Verify the Calculation

Double-check the calculation by performing the multiplication directly: \[21 \times 21 = 21 \times (20 + 1) = (21 \times 20) + (21 \times 1) = 420 + 21 = 441\] Ensure there are no calculation errors.

Key Concepts

Square Numbers

Square numbers are a foundational concept in prealgebra. They are special numbers that result from multiplying a number by itself. This operation is often denoted as "squaring" a number. For example, the square of a number like 5 is computed as \(5 \times 5 = 25\).
When you square a number, you are essentially creating a perfect square, which is a visual concept showing a square with equal sides. The area of such a square with side length 5, for example, would be 25 square units, as both sides are multiplied by each other.
Some properties of square numbers include:

• They are always non-negative, regardless of whether the original number is positive or negative.
• Square numbers end with digits 0, 1, 4, 5, 6, or 9. They never end with 2, 3, 7, or 8.

Understanding the concept of square numbers is not only vital for solving simple prealgebra problems but also for diving into more complex mathematical topics later on.

Multiplication

Multiplication is one of the four elementary arithmetic operations. It involves calculating the total of one number being added to itself a certain number of times. For instance, multiplying 3 by 4 is the same as adding 3 to itself 4 times: \(3 + 3 + 3 + 3 = 12\).
The process of multiplication is denoted with a multiplication sign (\( \times \)) or by simply juxtaposing numbers in parentheses, such as \(3 \cdot 4\) or \((3)(4)\).
Multiplication has several helpful properties:

• Commutative Property: The order of numbers does not change the result. For example, \(3 \times 4 = 4 \times 3\).
• Associative Property: The grouping of numbers can be changed. For example, \((2 \times 3) \times 4 = 2 \times (3 \times 4)\).
• Distributive Property: Allows you to break numbers down into parts. For example, \(21 \times 21 = 21 \times (20 + 1) = (21 \times 20) + (21 \times 1)\).

Understanding multiplication deeply enables you to compute more complex operations, such as squaring numbers or handling multi-digit products.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks of mathematics used in a wide range of contexts from day-to-day calculations to complex scientific equations.
Each operation serves a specific purpose:

• Addition results in a sum, combining two or more numbers.
• Subtraction finds the difference between numbers, essentially removing one number from another.
• Multiplication, as previously noted, calculates a total number when one is grouped a certain number of times.
• Division splits a number into equal parts or determines how many times one number is contained within another.

In the context of squaring a number like 21, these operations come together. Squaring combines multiplication and then addition through the distributive property, enabling calculations and simplifying operations. Mastery of arithmetic operations is essential for tackling all sorts of mathematical tasks efficiently.