If 2^(x+1)+2^x = 3^(y+2)-3^y, and x and y are integers, calculate the value of x + y ...................... edited

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Jeff1801

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I have tried to simplify this but the furthest I got was 3 * 2^x = 8 * 3^y. I'm not sure where to go from here.
 
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Jeff1801 said:
I have tried to simplify this but the furthest I got was 3.2^x = 8.3^y. I'm not sure where to go from here.
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How did you get

3.2^x = 8.3^y

Please share your work - so that we can help you properly.
 
Subhotosh Khan said:
How did you get

3.2^x = 8.3^y

Please share your work - so that we can help you properly.
Click to expand...
I simplified it like this.

2^(x+1)+2^x = 3^(y+2)-3^y
2^x(2+1) = 3^y(9-1)
(3)(2^x) = (8)(3^y)

I'm not sure how to go about getting x +y though. I can't seem to be able to manipulate it in the right way.
 
Last edited:
Jeff1801 said:
I simplified it like this.

2^(x+1)+2^x = 3^(y+2)-3y
2^x(2+1) = 3^y(9-1)
(3)(2^x) = (8)(3^y)

I'm not sure how to go about getting x +y though. I can't seem to be able to manipulate it in the right way.
Click to expand...
It should be:

2^(x+1)+2^x = 3^(y+2)-3y

2*2^x + 2^x = 3^2 * 3^y - 3 * y

3 * 2^x = 9 * 3^y - 3 * y
 
Jeff1801 said:
I have tried to simplify this but the furthest I got was 3.2^x = 8.3^y. I'm not sure where to go from here.
Click to expand...
I edited your OP.

Do you see at least one solution by observation?........ (remember 2 and 3 are relatively prime)
 
Jeff1801 said:
I have tried to simplify this but the furthest I got was 3 * 2^x = 8 * 3^y. I'm not sure where to go from here.
Click to expand...
Express everything in terms of the bases 2 and 3, with 2^something on one side and 3^something on the other. What must both be equal to?
 
3 * 2^x = 8 * 3^y gives 2^(x-3) = 3^(y-1)

What are the common value(s) that powers of 2 and powers of 3 have?
 
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