Consider this equations:
\(\dfrac{8^x}{2^{x+y}}=64\) and \(\dfrac{9^{x+y}}{3^{4y}}=243\)
Find the value of \(2xy\).
8^x 9^(x + y)
________ = 64 ________ = 243
2^(x + y) 3^(4y)
Write the first as Write the second as
(2^3)^x (3^2)^(x + y)
________ = 64 ____________ = 243
2^x * 2^y 3^(4y)
2^(3x) 3^(2x) * 3^(2y)
________ = 64 _____________ = 243
2^x * 2^y 3^(2y) * 3^( 2y)
2^(2x) 3^(2x)
_____ = 64 ________ = 243
2^y 3^(2y)
2^(2x - y) = 2^6 3^(2x - 2y) = 3^5
So
2x - y = 6 ⇒ -2x + y = -6 (1)
2x - 2y = 5 (2)
add (1) and (2)
-y = -1
y = 1
2x - 1 = 6
2x = 7
x = 7/2
2xy = 2(7/2)(1) = 7