If 3^(x + y) = 81 and 81^(x - y) = 9,then what is the value of the product xy? Express your answer as a common fraction.
+128460 3^(x + y) = 81
81^(x - y) = 9
Note that we can write this as
3^(x +y) = 3^4
(3^4)^(x - y) = 3^2 simplify
3^(x + y) = 3^4
3^( 4x - 4y) = 3^2
Equate exponents
x + y = 4 ⇒ y = 4 - x (1)
4x - 4y = 2 (2)
Sub (1) into (2)
4x - 4 ( 4 - x) = 2
4x - 16 + 4x = 2
8x = 18
x = 18 / 8 = 9/4
And
y = 4 - 9/4 = 7/4
So
x * y = (9/4) ( 7/4) = 63 / 16
cc
