Solving the Expression 2√3 2 - √3: A Comprehensive Guide

Whether you are a student or a professional who frequently deals with mathematical expressions, it is crucial to understand how to simplify expressions like 2√3 2 - √3. This article will provide a step-by-step process to break down and solve this expression, along with relevant solutions and explanations.

Step-by-Step Breakdown of 2√3 2 - √3

Let's start by breaking down the expression 2√3 2 - √3. In this expression, we are dealing with the combination of constants, square roots, and basic algebraic operations. Here's a detailed step-by-step guide on how to simplify it.

Combining Like Terms

The first approach to solving 2√3 2 - √3 is to combine the like terms. When we take a closer look, we can separate the expression into two parts:

2√3 2 - √3

The term 2√3 and -√3 can be combined as they are like terms involving the square root of 3. Let's see how this works.

Combining 2√3 and -√3

The term -√3 can be written as -1√3. Now, combining 2√3 and -1√3, we get:

2√3 - √3 (2 - 1)√3 √3

Substituting this back into the original expression, we have:

2√3 2 - √3 √3 2

Simplifying Further

Now, we can further simplify √3 2:

√3 2 2√3

Since there are no further like terms to combine, we conclude that the simplified form of the expression is:

2√3

Alternative Method: Algebraic Manipulation

Another way to approach the expression 2√3 2 - √3 is to manipulate it algebraically. Let's see the steps involved in this method:

Using the Square of a Difference

The expression 2√3 2 - √3 can also be viewed as the square of a difference:

(2 - √3)2 (2 - √3)(2 - √3) 22 - 2(2)(√3) (√3)2

Expanding this, we get:

4 - 4√3 3 7 - 4√3

To match the original expression, we take the positive root of this result:

√(7 - 4√3) √3

This results in:

7 - 4√3 3

Thus, the simplified form is:

2√3

Verification Using Approximate Values

To verify the results, we can use the approximate values for √3. It is known that √3 ≈ 1.73205. Let's substitute this value into the original expression:

Substituting √3 ≈ 1.73205

2√3 2 - √3 ≈ 2(1.73205) 2 - 1.73205