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)
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The following expression has to be simplified: 2^(1 - n)*(sqrt 8)^n)/(sqrt 2)^-n
2^(1 - n)*(sqrt 8)^n)/(sqrt 2)^-n
=> 2^(1 - n)*(2^(3/2)^n)/(2^(-n/2))
=> 2^(1 - n)*(2^(3n/2)/(2^(-n/2))
=> 2^(1 - n)*(2^(3n/2 + n/2)
=> 2^(1 - n)*(2^(2n))
=> 2^1*2^(2n - n)
=> 2^1*2^n
=> 2^(1+n)
The required result is option 4: 2^(n+1)
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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).
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