SOLUTION: n-th root of [32/{2^(5+n)}] I am having difficulties with this problem. Please guide me through it. Thank you.

Algebra ->  Exponents -> SOLUTION: n-th root of [32/{2^(5+n)}] I am having difficulties with this problem. Please guide me through it. Thank you.      Log On


   

Question 426357: n-th root of [32/{2^(5+n)}]
I am having difficulties with this problem. Please guide me through it. Thank you.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%28n%2C+32%2F2%5E%285%2Bn%29%29
The keys to this problem are:
  • Recognizing (or figuring out) that 2%5E5+=+32
  • Knowing that the rules for exponents work in both directions.

If we were to multiply 2%5E5%2A2%5En the rule for exponents would say to add the exponents. So
2%5E5%2A2%5En+=+2%5E%285%2Bn%29
It works backwards, too. Your denominator is the right side of the above equation. So we can replace your denominator with the left side:
root%28n%2C+32%2F%282%5E5%2A2%5En%29%29
Since 2%5E5+=+32 they cancel each other out in the fraction:
root%28n%2C+cross%2832%29%2F%28cross%282%5E5%29%2A2%5En%29%29
leaving:
root%28n%2C+1%2F2%5En%29
Now we can use a property of radicals, root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29, to split this radical:
root%28n%2C+1%29%2Froot%28n%2C+2%5En%29
And both of these nth roots simplify:
1/2