How do you write #625^(3/4)# in radical form?

2 Answers
Konstantinos Michailidis
May 28, 2016

Notice that #625=5^4# hence

#root 4 (625^3)=root 4 (5^12)=5^3=125#

Truong-Son N.
May 28, 2016

If you don't automatically know that #5^4 = 625#, or that #25^2 = 625#, another way to do this is:

#color(blue)(625^"3/4")#

#= (600 + 25)^"3/4"#

#= (60*10 + 25)^"3/4"#

#= (120*5 + 25)^"3/4"#

#= (12*50 + 25)^"3/4"#

#= (24*25 + 25)^"3/4"#

#= (25^2)^"3/4"#

Remember that #(x^a)^b = x^(a*b)#.

#= 25^(2*"3/4")#

#= 25^("6/4")#

#= 25^("3/2")#

#= 25^(3*"1/2")#

#= (5^2)^(3*"1/2")#

#= 5^(2*3*"1/2")#

Remember that multiplication is commutative.

#= 5^(3*2*"1/2")#

#= (5*5*5)^(2*"1/2")#

#= (25*5)^(1)#

#= color(blue)(125)#

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