+1245 If $3^{x + y} = 81$ and $81^{x - y} = 3,$ then what is the value of the product $xy$? Express your answer as a common fraction.
3^{x + y} = 81, 81^{x - y} = 3, solve for x, y
3^(x +y) =3^4.................(1)
3^(4x - 4y) =3^1.............(2)
Equate the exponents on both equations:
(x + y) =4.......................(3)
(4x - 4y) = 1...................(4)
From (3) above:
x = 4 - y sub this into (4) above
4(4 - y) - 4y = 1
16 - 4y - 4y = 1
16 - 8y = 1
15 - 8y = 0
15 = 8y
y = 15/8 =1 7/8, and x =17/8 =2 1/8
x*y =17/8 * 15/8 =255/64
