What is the square root of 27 to the power of 3?

1 Answer
Jul 17, 2015

Answer:

#sqrt(27)^3 = sqrt(27^3) = sqrt(3^9) = 3^(9/2) = 3^4 3^(1/2) = 81sqrt(3)#

Explanation:

Use the following identities (#a, b, c>= 0#):

#sqrt(a) = a^(1/2)#

#(a^b)^c = a^(bc)#

#a^(b+c) = a^b a^c#

Since the question is slightly ambiguous, let me first show that both possible meanings work out the same:

#sqrt(27)^3 = sqrt(27)sqrt(27)sqrt(27) = sqrt(27*27*27) = sqrt(27^3)#

Now #27 = 3^3#, so

#sqrt(27^3) = sqrt((3^3)^3) = sqrt(3^(3*3)) = sqrt(3^9)#

Then:

#sqrt(3^9) = (3^9)^(1/2) = 3^(9*1/2) = 3^(9/2) = 3^(4+1/2) = 3^4 3^(1/2) = 81sqrt(3)#

So: #sqrt(27)^3 = sqrt(27^3) = 81sqrt(3)#