Problem 10 - College Algebra

Question

Simplify each expression. $$\sqrt{1}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{1} \) is 1.

Step 1: Understand the Problem

The problem requires simplifying the expression \( \sqrt{1} \). Simplifying involves reducing the expression to its simplest form.

Step 2: Recall the Definition of Square Root

The square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). In other words, we are looking for a number that, when multiplied by itself, yields the original number inside the square root symbol.

Step 3: Apply the Definition to the Problem

We apply this concept to \( \sqrt{1} \). We need to find a number \( y \) such that \( y^2 = 1 \). The equation \( y^2 = 1 \) holds true when \( y = 1 \) or \( y = -1 \).

Step 4: Choose the Principal Square Root

By convention, the square root symbol \( \sqrt{} \) refers to the principal (or positive) square root. Hence, \( \sqrt{1} = 1 \).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we're referring to the process of reducing complicated expressions into a simpler or more manageable form. This often involves performing calculations, removing parentheses, and combining like terms. In the case of square roots, simplifying an expression means finding the simplest value of the root.

To simplify an expression like \( \sqrt{1} \), follow these steps:

• Identify the expression you want to simplify. In this example, it’s \( \sqrt{1} \).
• Recall any relevant mathematical rules or definitions. For square roots, this includes knowing what square root means, which will help you simplify correctly.
• Determine the simplest form of the expression based on these rules. In this case, recognizing that the principal square root of 1 is simply 1.

Simplifying not only helps solve equations but also aids in understanding the relationships between numbers better. It might not seem needed when the answer is obvious, like \( \sqrt{1} = 1 \), but it's a useful skill for more complicated expressions.

Principal Square Root

Understanding what a principal square root is will make it much clearer why the square root of a number often returns only one outcome. Every positive real number has two square roots: one positive and one negative. The principal square root is the non-negative square root.

Take the concept of \( \sqrt{1} \) as an example. Both 1 and -1 satisfy the condition of multiplying by themselves to get 1. But when using the square root symbol \( \sqrt{} \), it refers specifically to the principal square root. Hence,

• The principal square root of 1 is 1, not -1.
• Similarly, the principal square root of 4 is 2, not -2.

This convention makes calculations straightforward and consistent, particularly in algebra and calculus, where the direction of equations often matters. Choosing the principal root ensures there is only one standardised answer whenever you solve square root problems.

Definition of Square Root

Before diving into square root problems, it is crucial to understand what a square root actually is. In simple terms, the square root of a number \( x \) is a value \( y \) which, when multiplied by itself, returns \( x \).

So for the expression \( \sqrt{1} \):

• You are searching for a number \( y \) such that \( y^2 = 1 \).
• Both numbers 1 and -1 fulfill the condition of being squared to result in 1, since \( 1^2 = 1 \) and \( (-1)^2 = 1 \).

The definition helps clarify why square root results have two potential factors, one positive and one negative. However, thanks to mathematical conventions prioritizing the principal square root, we often default to the positive root providing a consistent framework for further mathematical exploration.