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

sqrt((3^15))

1 Answer
Jan 27, 2018

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

Explanation:

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 15th power, i.e. (sqrt(3))^15

  • Raise 3 to the 15th 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)