SOLUTION: give the principal roots of each of the following.(for sqaure root is required unless the problem indicates otherwise.) 5. square root of 49 6.square root of -27 cubed

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Question 252903: give the principal roots of each of the following.(for sqaure root is required unless the problem indicates otherwise.)
5. square root of 49
6.square root of -27 cubed
Found 3 solutions by Greenfinch, drk, Theo:
Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
5. +or - 7
6. -27 = {(-3)^3}^3
= {-3}^9
so sqrt is {-3}^(9/2)
+ or - 81 x (sqrt3) x i

Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
5sqrt%2849%29
the sqrt%2849%29 = 7, so
we get 5x7 = 35.
--
sqrt%28-27%29%5E3
%28-27%29%5E3+=+%28%28-3%29%5E3%29%5E3+=+%28-3%29%5E9
So,
sqrt%28%28-3%29%5E9%29.
I don't know what you know about negative square roots: We have one of two answers:
(i) no solution.
sqrt%28%283%29%5E9%29+%2A+sqrt%28%28-1%29%5E9%29
81sqrt(3) x i
(ii) 80isqrt(3)

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2849%29 = sqrt%287%2A7%29 = 7.

sqrt%28-27%29%5E3 = %28sqrt%283%2A3%2A-3%29%29%5E3 = %283%2Asqrt%28-3%29%29%5E3 = 3%5E3+%2A+sqrt%28-3%29%5E3 = 27%2Asqrt%28-3%29%5E2%2Asqrt%28-3%29 = 27%2A-3%2Asqrt%28-3%29 = -81%2Asqrt%28-3%29

to confirm first equation:

sqrt%287%5E2%29 = sqrt%2849%29 = 7.

to confirm second equation:

sqrt%28%28-81%2Asqrt%28-3%29%29%5E2%29 = sqrt%28%28-81%29%5E2%2Asqrt%28-3%29%5E2%29 = sqrt%286561%2A%28-3%29%29 = sqrt%28-19683%29.

sqrt%28-19683%29 = sqrt%28%28-27%29%5E3%29

sqrt%28%28-27%29%5E3%29 is the same as %28sqrt%28-27%29%29%5E3.

this can be seen easier with exponential notation.


sqrt%28%28-27%29%5E3%29 is the same as %28%28-27%29%5E3%29%5E%281%2F2%29 which equals %28-27%29%5E%283%2A.5%29 which equals %28-27%29%5E%281.5%29.


sqrt%28-27%29%5E3 is the same as %28%28-27%29%5E%281%2F2%29%29%5E3%29 which equals %28-27%29%5E%28.5%2A3%29 which equals %28-27%29%5E%281.5%29.