+159 Solve for the rational numbers x and y:
\(2^{x+y} \cdot 3^{x-y} \cdot 6^{2x+2y}= 72\)
Show answer in ordered pair. (x, y)
+2863 Prime factorize
72
8 * 9
23 * 32
Factor out a 62
2 * 62
Make system of equations
2x + 2y = 2
x + y = 1
Solving, we get x = 1/2 and y = 1/2
+129907 2^(x + y) * 3^(x - y) * 6^(2x + 2y) = 72
2^(x + y) * 3^(x - y) * ( 6^2)^(x + y) = 72
2^(x + y) * 3(x - y) * 36(^(x + y) = 72
( 2 * 36)^(x + y) * 3^(x - y) = 72
(72)^(x + y) * 3^(x - y) = 72
Since
72^(1) * 3^(0) = 72
This will be true when
x + y = 1
x - y = 0 add these
2x = 1
x =1/2
And
1/2 + y = 1
y = 1/2
(x,y) = (1/2, 1/2)
