Simplifying the Expression 2 / (2 - √18): A Comprehensive Guide for SEO

Introduction

Rationalizing expressions involving surds can be a tricky process, but with the right techniques, it becomes much more manageable. This article provides a detailed step-by-step guide on how to simplify the expression 2 / (2 - √18). We will break down the process into manageable steps, ensuring each step is clear and understandable, making it easier for students and professionals alike to grasp the concepts.

Simplifying the Square Root Term

The first step in simplifying 2 / (2 - √18) involves simplifying the square root term √18. We start by expressing 18 as a product of its prime factors:

Step 1: Decomposing the Square Root

18 can be written as 9 × 2. Since 9 is a perfect square, we can simplify √18 as follows:

√18 √(9 × 2) √9 × √2 3√2

Reforming the Expression

Once we have simplified √18 to 3√2, we can rewrite the original expression as:

2 / (2 - 3√2)

Step 2: Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (2 - 3√2) is (2 3√2).

2 / (2 - 3√2) × (2 3√2) / (2 3√2) [2(2 3√2)] / [(2 - 3√2)(2 3√2)]

Calculating the Denominator

The denominator, (2 - 3√2)(2 3√2), can be simplified using the difference of squares formula:

(2 - 3√2)(2 3√2) 22 - (3√2)2 4 - 18 -14

Calculating the Numerator

The numerator, 2(2 3√2), is calculated as follows:

2(2 3√2) 2 × 2 2 × 3√2 4 6√2

Combining the Results

Now, we can combine the results into a single fraction:

(4 6√2) / -14

Step 3: Simplifying the Fraction

Finally, we can simplify the fraction by dividing both terms in the numerator by -14:

(4 6√2) / -14 -2 / 7 - 3√2 / 7

Final Answer

The simplified form of the expression 2 / (2 - √18) is:

-2 / 7 - 3√2 / 7

Conclusion

By following these detailed steps, we have successfully simplified the expression involving a surd. This process can be applied to other similar expressions, providing a robust framework for solving more complex algebraic problems. If you are looking for more resources on simplifying surds or related algebraic expressions, this article can be a valuable guide.

Related Keywords

Keyword 1: Simplify surd

Keyword 2: Rationalize denominator

Keyword 3: Algebraic simplification