Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{1}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Understand the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For this exercise, we need to determine what number multiplied by itself equals 1.

Step 2: Identify the Perfect Square

A perfect square is a number that is the square of an integer. Since \(1 \times 1 = 1\), the number 1 is a perfect square.

Step 3: Simplify the Square Root

Given \(\sqrt{1}\), we replace it with the integer whose square is 1. Since \(1^2 = 1\), we can simplify \(\sqrt{1}\) to 1. The answer is simply \(1\).

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. Recognizing these numbers helps simplify expressions involving square roots. For example, the integer 2 gives the perfect square 4 since \(2 \times 2 = 4\).

• Examples of perfect squares include 1, 4, 9, 16, and 25.
• This shows that \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), and so on.

Knowing this pattern is crucial for identifying numbers whose square roots are integers. When you see a number under a square root, check if it's a perfect square. If it is, you can easily find the square root by reversing the operation and finding the original integer. Simplifying square roots becomes straightforward when the number inside the root is a perfect square.

Basic Algebra Skills

Simplifying square roots is a skill that relies on your understanding of basic algebra concepts. Algebra often involves manipulating expressions to make them simpler. A key part of this is recognizing numerical patterns and relationships, such as perfect squares.

• Square roots undo the operation of squaring a number.
• When simplifying, always look to match the root with a perfect number.
• Use operations and transformations to express the simplest form.

In exercises like simplifying \(\sqrt{1}\), you identify that 1 is a perfect square. Then use basic algebra principles to show that its square root is 1. This process sharpens overall algebra skills by enhancing your pattern-recognition abilities. By practicing, you become more adept at transforming complex expressions into simpler forms using algebraic methods.

Square Root Properties

Square roots possess several properties that aid in simplifying them and understanding mathematical relationships. Grasping these properties can make tasks like simplifying square root expressions much easier.

• The square root of a perfect square is an integer. This is because a perfect square results from an integer multiplied by itself.
• Knowing that \(\sqrt{a} \times \sqrt{a} = a\) helps solve problems efficiently.
• The square root function is non-negative, meaning \(\sqrt{x} \geq 0\) for all non-negative x.

Applying these properties is simple. If you have \(\sqrt{1}\), you recognize that 1 is already a perfect square. Its square root is just the integer 1. These traits of square roots simplify your work and help make sense of mathematical operations effectively.