If n >0 and 9 = n^6, then 27n is equal to all of the follwoing except:
a) 81/n
b) 3n^7
c) 9n^4
d) n^10
e) n^13/3
To find the expressions that are equal to 27n, we can first solve the equation 9 = n^6.
Taking the sixth root of both sides, we have:
n^6 = 9
n = 9^(1/6)
Now let's substitute this value of n into the answer choices to check which ones are equal to 27n:
a) 81/n = 81 / (9^(1/6)) = (9^2) / (9^(1/6)) = 9^(12/6 - 1/6) = 9^(11/6)
b) 3n^7 = 3(9^(1/6))^7 = 3 * 9^(7/6) = 3 * (9^(6/6) * 9^(1/6)) = 3 * (9 * 9^(1/6)) = 3 * (9^(7/6 + 1)) = 3 * 9^(13/6)
c) 9n^4 = 9(9^(1/6))^4 = 9 * 9^(4/6) = 9 * 9^(2/3) = 9^(1/6 + 2/3) = 9^(5/6)
d) n^10 = (9^(1/6))^10 = 9^(10/6) = 9^(5/3)
e) n^13/3 = (9^(1/6))^(13/3) = 9^(13/18)
So, the expression 27n is equal to a), b), and c), but it is not equal to d) or e).
Therefore, the correct answer is:
d) n^10