SOLUTION: (2^n-3)*(8^n+1)/(2^2n-1)*(4^2-n)

Algebra ->  Exponents -> SOLUTION: (2^n-3)*(8^n+1)/(2^2n-1)*(4^2-n)      Log On


   

Question 421404: (2^n-3)*(8^n+1)/(2^2n-1)*(4^2-n)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I assume the expression is:

If this is correct then please put parentheses around exponents. Also put parentheses around numerators and denominators. The way you posted the expression meant:
%282%5En-3%29%2A%288%5En%2B1%29%2F%282%5E2n-1%29%2A%284%5E2-n%29
which is not the same thing!

Part of simplifying this expression requires that we recognize that the bases of 8 and 4 are powers of 2. 8+=+2%5E3 and 4+=+2%5E2. This means that all the bases can be 2's:

Now we can start simplifying. Exponents first. The rule for exponents when raising a power to a power is to multiply the exponents. Using this rule we get:

which simplifies to:
%282%5E%28n-3%29%2A2%5E%283n%2B3%29%29%2F%282%5E%282n-1%29%2A2%5E%284-2n%29%29
Next is multiplication. The rule for exponents when multiplying (when the bases are the same) is to add the exponents. Using this rule in both the numerator and denominator we get:
2%5E%28%28n-3%29%2B%283n%2B3%29%29%2F2%5E%28%282n-1%29%2B%284-2n%29%29
which simplifies to:
2%5E%284n%29%2F2%5E%283%29
From here we can go in two directions. One, we could use the rule for exponents when dividing, i.e. subtract the exponents:
2%5E%284n-3%29
This is one possible answer.

Another direction we could go with the 2%5E%284n%29%2F2%5E%283%29 expression is to take advantage of the fact that there is no variable in the denominator's exponent. And without a variable we can find the actual value for the denominator. Since 2%5E3+=+8
our expression becomes:
2%5E%284n%29%2F8
This is another, correct answer to this problem.

We could also take the last answer a step further:
2%5E%284n%29%2F8
%28%282%5E4%29%5En%29%2F8
16%5En%29%2F8

There is no one "perfectly correct" answer for this problem. All three answers we found are correct and a case can be made for each one being "the best" answer:
2%5E%284n-3%29
2%5E%284n%29%2F8
16%5En%29%2F8