How do you simplify square root of three to the 15th power?

Question

#sqrt((3^15))#

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Answer 1 · 2018-01-27T00:36:49

#sqrt(3^15) = (sqrt(3))^15 = 2187 sqrt(3)#

The question is slightly ambiguous in that it could mean either of the following:

Take the square root of #3# then raise it to the #15#th power, i.e. #(sqrt(3))^15#
Raise #3# to the #15#th power then take the square root, i.e. #sqrt(3^15)#

In general if #a >= 0# then #sqrt(a^2 b) = a sqrt(b)#

So we find:

#sqrt(3^15) = sqrt((3^7)^2 * 3) = 3^7 sqrt(3) = 2187 sqrt(3)#

Also:

#(sqrt(3))^15 = (sqrt(3))^14 sqrt(3) = ((sqrt(3))^2)^7 * sqrt(3) = 3^7 sqrt(3) = 2187 sqrt(3)#