(2^k+1)+(2^k+1)====(2^k+2) and why not (2^2k+2) ............

Please explain this how and which formula used to solve this..............

and what about this equation (2^k+1)+(3^k+1).............Please explain differently

2^(k + 1) + 2^(k + 1) = 2[2^(k + 1)]
= 2^1 x 2^(k + 1)
= 2^[1 + k + 1]
= 2^(k + 2)

rules of exponents...

remember what 2^(k + 1) means

let k + 1 = 3
so you have 2^3 = 2 x 2 x 2

and 2^3 + 2^3 = (2 x 2 x 2) + (2 x 2 x 2)
= 2(2 x 2 x 2) = 2 x 2 x 2 x 2 = 2^4

*****
you can't simplify 2^(k + 1) + 3^(k + 1)

these aren't like terms, thus they can't be combined by addition
(and don't try to make it (2 + 3)^(k + 1) )

x^3 + y^3 =/= (x + y)^3

just like you can't say that 3^2 + 4^2 = (3 + 4)^2

Hello MathQuestion, you recall a+a = 2a
Here a = 2^(k+1)
So 2^(k+1) + 2^(k+1) = 2 * 2^(k+1)
Recall m^2 * m^3 = m^ (2+3) = m^5
So 2^1 * 2^(k+1) = 2^(1+k+1) = 2^ (k+2)
If it has to be 2^(2k+2) then instead adding you have to multiply them
So 2^(k+1) * 2^(k+1) = 2^(2k+2)
In the next one we cannot simplify further. In both terms the base are different.
So (2^k+1)+(3^k+1) remains as the same without further simplification.