Indices question. Simplify?

If the exponent is n+1, it needs parentheses: 2^(n+1)

(2ⁿ⁺¹×2ⁿ)/2ⁿ = 2ⁿ⁺¹×(2ⁿ/2ⁿ) = 2ⁿ⁺¹×1 = 2ⁿ⁺¹
:::::
(2ⁿ⁺¹+2ⁿ)/2ⁿ = 2ⁿ⁺¹/2ⁿ + 2ⁿ/2ⁿ = 2ⁿ⁺¹⁻ⁿ + 1 = 2¹ + 1 = 3

First, a little notation. In algebra, x is very often a variable, which is a major reason that most books use a dot, writing that first expression as something like (2ⁿ • 2ⁿ⁺¹) / 2ⁿ. That's hard to do on a keyboard (and only barely possible for that expression using the few superscript characters that are available.) So, when typing, many people use * for that dot and--just as you did--also use ^ for powers. However, the ^ doesn't actually raise and shrink anything to the right, so if you want an expression as an index, you need parentheses. As in [2^(n+1) * 2*n] / 2^n

Now to the math...Use the property a^b * a^c = a^(b+c) to rewrite the numerator:

[2^(n+1) * 2*n] / 2^n = 2^(n + 1 + n) / 2^n
= 2^(2n + 1) / 2^n .... then simplify
= 2^(2n + 1 - n) .... now use the property a^b / a^c = a^(b-c)
= 2^(n + 1) .... and simplify

In the second expression, you can start with the distributive property, (a + b)/c = a/c + b/c

[2^(n + 1) + 2^n) / 2^n = 2^(n + 1) / 2^n + 2^n / 2^n
= 2^(n + 1 - n) + 1 .... Use a^b/a^c = a^(b-c) again, and anything (nonzero) over itself is 1
= 2^1 + 1 .... simplify
= 2 + 1
= 3