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.
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.
34 = 81 so (x+y) must equal 4
81–2 = 9 so (x–y) must equal –2
x + y = 4
x – y = –2
Add those two equations and obtain 2x = 2 therefore x =1
Sub 2 back into one of the equations 1 + y = 4 therefore y=3
x • y = 1 • 3 = 3
I suppose, to comply with the terms of the 3
problem, the answer should be expressed xy = ––
as a fraction. 1
3^(x +y) = 81
81^(x - y) = 9
Write everything in terms of base 3
3^(x +y) = 3^(4)
3^[4(x -y)] = 3^(2)
Equate the exponents ......we have these equations
x + y = 4
4 [ x - y] = 2 → 2x - 2y = 1 → x - y =1/2
x + y = 4
x - y =1/2 add these equations
2x = 9/2
x = 9/4
And
x + y = 4
9/4 + y =16/4
y = 7/4
xy = (9/4)(7/4) = 63/16