Which if the following is the simplified form of 2^(1-n)*(sqrt 8)^n/ (sqrt 2)^(-n).Options: 1. 2^n 2. 2^(-n) 3. 2^(n-1) 4. 2^(n+1)

First of all, we'll write sqrt8 as a power of
2:

sqrt8 = (8)^(n/2) = (2^3)^(n/2) =
2^(3n/2)

Now, we'll multiply 2^(3n/2) by
2^(1-n).

Since the bases are matching, we'll add the
exponents:

2^(3n/2) *2^(1-n) = 2^(3n/2 + 1 - n) = 2^(n/2 +
1)

Now, we'll write the denominator as a power of
2:

(sqrt2)^(-n) =
2^(-n/2)

Now, we'll re-write the
ratio:

2^(n/2 + 1)/2^(-n/2) = 2^(n/2 + 1+
n/2)

2^(n/2 + 1+ n/2) = 2^(2n/2 +
1)

 2^(2n/2 + 1) =
2^(n+1)

The correct option is the 4th:
2^(n+1).