Exploring the Value of √x / ?√x for Given Values of x

This article aims to explore the values of the algebraic expression √x / √x for specific values of x. This is a classic problem in algebra and can help us understand the properties of square roots and fractions involving square roots. The procedures and techniques used in these explorations will be detailed, illustrated with examples, and verified using algebraic properties and rules.

Case I: x 32√2

Let's begin with the expression given: x 32√2. Rewriting the expression: x 32√2 12√2 times; 2 x 121√2 times; √2^2 1√2^2 This simplifies to: √x ±√2 times; 1 ±√2 Further simplification: x / √x (32√2) / (1√2) 2√2 Similarly, when √x -√2, we find: x / √x (32√2) / (-√2) -2√2

Case II: Further Exploration with x 32√2

Let's further explore the expression x / √x for x 32√2 using the same operations: Considering the simplified form: x 1√2^2 Then, we have: √x / √x (1√2) / (1√2) 1 From the above, it's clear that: x / √x (32√2) / (1√2) 2√2

Case III: A General Approach to √x / √x

Let's derive a more generalized form for the expression given the condition 32√2 - 2√2 1:

Starting with the expression: √x / √x √32√2 / √32√2 √32√2 times; (√3 - 2√2) / √32√2 times; (√3 - 2√2) The square of the expression: (√32√2 times; (√3 - 2√2))^2 32√2 times; (3 - 2√2) 8 This simplifies to: √32√2 times; (√3 - 2√2) √8 2√2

Conclusion

These explorations and calculations demonstrate the power of algebraic manipulation and the properties of square roots. With the given values and expressions, we were able to find the values of √x / √x through substitution, simplification, and rationalization.