Let's solve the problem step-by-step to find the value of [tex]\( x^2 + y^2 \)[/tex], given the expressions for [tex]\( x \)[/tex] and [tex]\( y^2 \)[/tex].
Given:
[tex]\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \][/tex]
[tex]\[ y^2 = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \][/tex]
First, find the value of [tex]\( x \)[/tex]:
[tex]\[ x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \][/tex]
To evaluate [tex]\( x \)[/tex]:
[tex]\[ x \approx 9.898979485566363 \][/tex]
Next, find the value of [tex]\( y \)[/tex] from [tex]\( y^2 \)[/tex]:
[tex]\[ y^2 = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \][/tex]
To evaluate [tex]\( y \)[/tex]:
[tex]\[ y^2 \approx 0.10102051443364375 \][/tex]
[tex]\[ y \approx 0.31783724519578216 \][/tex]
Now, compute [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]:
[tex]\[ x^2 = (9.898979485566363)^2 \approx 97.98979485566369 \][/tex]
[tex]\[ y^2 = (0.31783724519578216)^2 = y^2 \][/tex]
[tex]\[ y^2 \approx 0.10102051443364375 \][/tex]
Finally, find the sum:
[tex]\[ x^2 + y^2 = 97.98979485566369 + 0.10102051443364375 \approx 98.09081537009733 \][/tex]
So, the value of [tex]\( x^2 + y^2 \)[/tex] is:
[tex]\[ \boxed{98.09081537009733} \][/tex]
