If x and y are rational numbers such that 15+¡Ì2/15-¡Ì2 = x+y¡Ì2, then the value of x and y ate respectively:
To solve this problem, we need to rationalize the given expression and then compare the real and imaginary parts. Let's break it down step by step:
Step 1: Rationalizing the denominator:
We have the expression 15+¡Ì2/15-¡Ì2.
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is 15+¡Ì2.
This gives us:
(15+¡Ì2)(15+¡Ì2) / (15-¡Ì2)(15+¡Ì2)
Simplifying the numerator and denominator:
The numerator becomes (15*15) + (15*¡Ì2) + (¡Ì2*15) + (¡Ì2*¡Ì2)
The denominator becomes (15*15) + (15*¡Ì2) - (¡Ì2*15) - (¡Ì2*¡Ì2)
Simplifying further:
The numerator becomes 225 + 30¡Ì2 + 30¡Ì2 - 2
The denominator becomes 225 - 30¡Ì2 + 30¡Ì2 - 2
Simplifying more:
The numerator becomes 223 + 60¡Ì2
The denominator becomes 223
Step 2: Separate the real and imaginary parts:
Since we have the equation x + y¡Ì2, we need to compare the real parts and imaginary parts separately.
For the real parts:
On the left side, we have 223.
On the right side, we have x.
Thus, x = 223.
For the imaginary parts:
On the left side, we have 60¡Ì2.
On the right side, we have y¡Ì2.
Thus, y = 60.
So, the values of x and y are 223 and 60, respectively.