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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.

 Apr 12, 2021
 #1
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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

 

 

coolccoolccool

 Apr 12, 2021
edited by CPhill  Apr 12, 2021
edited by CPhill  Apr 12, 2021

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