Problem 47 - Prealgebra

Question

Use the following information. Squaring a number and finding the square root of a number are inverse operations. That is, one operation undoes the other operation. Use inverse operations to evaluate each expression. $$(\sqrt{100})^{2}$$

Step-by-Step Solution

Answer

The expression \((\sqrt{100})^2\) evaluates to 100.

Step 1: Evaluate the Square Root

Find the square root of 100. The square root function looks for a number that, when multiplied by itself, gives the original number. Here, \( \sqrt{100} = 10 \).

Step 2: Apply the Square Operation

Now square the result from Step 1. Squaring a number means multiplying it by itself. So, we calculate \( (10)^2 = 10 \times 10 = 100 \).

Step 3: Confirm Inverse Operation

Verify if the operations are truly inverse by comparing the final result with the original number under the square root. The initial expression was \( (\sqrt{100})^2 \), which simplifies to 100, confirming that squaring and finding the square root are inverse operations.

Key Concepts

Squaring a Number

Squaring a number is a simple yet powerful mathematical operation. It involves multiplying a number by itself. For example, if you have the number 5, squaring it would mean calculating \( 5 \times 5 \), which equals 25. The result of this operation is called a "square" of the original number, hence the name "squaring."

• Operation: Let’s say \( n \) is any number. Squaring it means \( n^2 = n \times n \).
• Usage: Squaring is often used in geometry, particularly when dealing with areas. For instance, the area of a square is calculated as the side length squared.

It’s vital to understand that squaring turns positive numbers and negative numbers into positive results. For instance, both \((4)^2\) and \((-4)^2\) yield 16. This property is key in solving equations and understanding functions that involve square terms.

Square Root

The square root operation is the reverse process of squaring a number. It seeks the original number whose square gives a specified number. When you see a square root symbol \( \sqrt{} \), it means you should find a number that, when squared, equals the number inside the radical. For example, the square root of 100 is 10 because \( 10 \times 10 = 100 \).

• Symbol: The square root is denoted by \( \sqrt{} \), and finding \( \sqrt{n} \) means resolving the equation \( x^2 = n \).
• Important Concept: Square roots pair with squaring as inverse operations. They effectively cancel eachother out. So, \( (\sqrt{100})^2 \) reduces back to 100.

When handling square roots, it’s important to consider both the positive and negative roots. However, in common arithmetic, the principal (positive) root is usually used unless otherwise specified.

Prealgebra

Prealgebra lays the foundation for algebra, introducing basic mathematical concepts and operations. It involves understanding numbers, operations, and relationships, preparing students for the complexity of algebra that follows.

• Core Ideas: Topics covered include arithmetic operations, fractions, decimals, and basics of geometry. Understanding the properties of operations, like the distributive, commutative, and associative properties, are essential.
• Importance: Prealgebra is crucial for grasping more advanced math. Without a solid foundation, algebra can seem daunting. This stage ensures students are comfortable with calculations and concepts.
• Skills Developed: Problem-solving, logical thinking, and reasoning are enhanced through exercises in prealgebra.

Grasping concepts like squaring numbers and finding square roots are part of prealgebra. These skills are integrated into solving equations and understanding functions, serving as building blocks for future math courses.