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.

Lightning Aug 2, 2018

#1**+1 **

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**

Guest Aug 2, 2018

edited by
Guest
Aug 2, 2018