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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.

 Aug 2, 2018
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  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

 Aug 2, 2018
edited by Guest  Aug 2, 2018

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