Understanding the Value of x and 1/x When x √3 √2

This article delves into the algebraic manipulation and the properties of square roots to find the value of x and its reciprocal, 1/x, when x is set to the product of √3 and √2.

Introduction to the Problem

Let x √3 √2. Our task is to determine the value of x and 1/x. This problem involves the manipulation of square roots and the algebraic identity (a - b) (a2 - b2). Understanding these concepts will help us solve the problem step by step.

Solving for x and 1/x

Given that x √3 √2, let's proceed to find 1/x and then combine the results.

Step 1: Expressing 1/x

First, we need to express 1/x. This can be written as:

1/x 1/√3 √2

To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator, which is (√3 - √2). This will help us eliminate the square root in the denominator.

1/x (√3 - √2) / ( (√3 √2) ( (√3 - √2) ) )

Using the algebraic identity (a - b) (a2 - b2), we simplify the denominator:

1/x (√3 - √2) / (3 - 2) (√3 - √2) / 1 √3 - √2

Thus, we have:

1/x √3 - √2

Step 2: Combining x and 1/x

Now that we have the values for x and 1/x, we can combine them. We can write:

x 1/x √3 √2 (√3 - √2)

x 1/x √3 √2 √3 - √2

Since √3 √2 √3 - √2 simplifies to:

2 √3

We can conclude that:

x 1/x 2 √3

Conclusion

In summary, we have determined that when x √3 √2, the value of 1/x is √3 - √2, and the sum of x and 1/x is 2 √3. These steps demonstrate the application of algebraic identities and the manipulation of square roots to solve complex problems.

Related Topics and Keywords

Keywords: x value, reciprocal, algebraic manipulation, square roots

Related Topics: Algebra, Square Roots, Rationalizing Denominators, Algebraic Identities