What is 2^x+2^y=40,2^{x+y}=256 ?

The solution to 2^x+2^y=40,2^{x+y}=256 is

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2^x+2^y=40,2^{x+y}=256

2^{x} + 2^{y} = 40, 2^{x + y} = 256

(x = 5, x = 3, y = 3 y = 5)

Solution steps

[2^{x} + 2^{y} = 40 2^{x + y} = 256]

Isolate

x

for

2^{x} + 2^{y} = 40 : x = \frac{ln ( 40 - 2 ^{y} )}{ln ( 2 )}

Plug the solutions

x = \frac{ln ( 40 - 2 ^{y} )}{ln ( 2 )}

into

2^{x + y} = 256

For

2^{x + y} = 256

, subsitute

x

with

\frac{ln ( 40 - 2 ^{y} )}{ln ( 2 )} : y = 3, y = 5

Plug the solutions

y = 3, y = 5

into

2^{x} + 2^{y} = 40

For

2^{x} + 2^{y} = 40

, subsitute

y

with

3 : x = 5

For

2^{x} + 2^{y} = 40

, subsitute

y

with

5 : x = 3

Verify solutions by plugging them into the original equations

Therefore, the final solutions for

2^{x} + 2^{y} = 40, 2^{x + y} = 256

are

(x = 5, x = 3, y = 3 y = 5)

a = \frac{2}{3} b, c = 12

4 x^{2} - 9 (y + 1)^{2} = 36, 2 x + 3 y = - 5

x^{2} - 25 = 0, y^{3} = 216

y^{2} = 89, x = 88

y x + 6 = 9, x + y = 6