Math Olympiad | A Nice Algebra Problem | Find the Value of n, x and y

[@hariomdas270]

✓y^n=13-7=6
✓x^n=7
x^n=49
y^n=36
n=2

[@satrajitghosh8162]

Writing u = x ^ (n/2), v = y ^ ( n /2) one gets
u ^2 - v ^ 2 = 13 = u + v
u - v = 1
Hereby x ^ (n/2) = u = 7
y ^ (n/2) = v = 6
x = 7 ^ ( 2/n ) = 7 , n = 2
y = 6 ^ ( 2/n ) = 6 , n = 2
Hereby n = 2, x = 7, y = 6

[@ivansouza6468]

√x^n + √y^n = 13
x^n - y^n = 13
x^n = (√x^n)^2
y^n = (√y^n)^2
(√x^n +√y^n).(√x^n - √y^n) = 23
13(√x^n - √y^n) = 13
√x^n - √y^n = 1
√x^n + √y^n = 13
2√x^n = 14
√x^n = 7
(√x^n)^2 = 7^2
x^n = 7^2 > x = 7 e n = 2
√7^2 + √y^n = 13
7 + √y^2 = 13
√y^2 = 13 - 7
√y^2 = 6
y = 6