X^K+2 split up exponent rules

Question

I am working on a problem and I am confused if exponents can be split up in the manner below.

$$x^{k+2} = x^{k+1}+x^{k+1} = 2x^{k+1} $$

I apologize if this is a simple question I tried to look up exponent rules but I couldn't find exponents + a number.

Answer 1

No, they definitely cannot be split up like that. To my knowledge there isn't any general law for $x^a + x^b = \text{something}$.

A few handy exponent laws can be found at http://mathworld.wolfram.com/ExponentLaws.html

Answer 2

No, they’re not broken up like that. You need to use

$$a^b\cdot a^c = a^{b+c}$$

and vice-versa. In general, you can try to create a simple expression to see if your way is correct. For example, notice

$$2^{2+2} = 16 \color{red}{\neq 2^2+2^2 = 8}$$