How do you find the square root of 5 to the 4th power?

2 Answers

It is #sqrt(5^4)=5^2=25#

Sep 14, 2015

Answer:

The question is slightly ambiguous, but both #(sqrt(5))^4# and #sqrt(5^4)# are equal to #25#

Explanation:

#(a^b)^c = a^(bc)# for #a, b, c >= 0#

So:
#(sqrt(5))^4 = ((sqrt(5))^2)^2 = 5^2 = 25#

Also:
#sqrt(5^4) = sqrt((5^2)^2) = 5^2 = 25#

Alternatively using fractional exponents:

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

So:
#(sqrt(5))^4 = (5^(1/2))^4 = 5^(1/2 * 4) = 5^2 = 25#

and:
#sqrt(5^4) = (5^4)^(1/2) = 5^(4*1/2) = 5^2 = 25#