This problem involves simplifying an expression with radicals. We are given that x = √13 + 2√3, and we need to find the value of x - 1/x. This type of problem requires a combination of algebraic manipulation and knowledge of radical properties....
Key Concepts
To solve this problem, we need to utilize the following key concepts:
- Rationalizing the Denominator: We need to get rid of the radical in the denominator of 1/x.
- Difference of Squares: We will use the formula (a + b)(a - b) = a² - b² to simplify the expression.
- Simplifying Radicals: We will need to simplify radical expressions to obtain a final solution.
Step-by-Step Solution
- Rationalizing the Denominator:
To rationalize the denominator of 1/x, we multiply both numerator and denominator by the conjugate of x, which is √13 - 2√3:
1/x = 1/(√13 + 2√3) * (√13 - 2√3)/(√13 - 2√3)
This simplifies to:
1/x = (√13 - 2√3)/(13 - 12) = √13 - 2√3
- Substituting and Simplifying:
Now, we can substitute the value of x and the simplified value of 1/x into the expression x - 1/x:
x - 1/x = (√13 + 2√3) - (√13 - 2√3)
Simplifying further, we get:
x - 1/x = √13 + 2√3 - √13 + 2√3 = 4√3
Final Answer
Therefore, the value of x - 1/x, when x = √13 + 2√3, is 4√3.
Conclusion
This problem highlights how understanding radical properties and algebraic techniques can help us simplify complex expressions. The key is to recognize patterns and apply the appropriate methods to reduce the expression to a simpler form. Rationalizing the denominator and utilizing the difference of squares are two powerful techniques that are frequently used in algebra.