How do you simplify #8^(2/3)#?

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Answer 1 · 2016-07-24T20:03:06

#=4#

#8^(2/3)# can be written as
#root3(8^2#
#=root3(64)#
#=root3((4)(4)(4)#
#=4#

Answer 2 · 2017-06-30T07:07:59

#4#

One of the laws of indices states: #x^(m/n) = rootn((x^m)) = (rootn(x))^m#

Therefore #8^(2/3)# can be written as #root3((8^2)) or (root3(8))^2#

I prefer to use the second form because it uses smaller numbers - the root is found first and then that is squared.

#root3(8) = 2 and 2^2 =4#

So: #(color(blue)(root3(8)))^2 = color(blue)(2)^2 = 4#

Consider a question such as #32^(3/5)#

#root5(color(blue)((32^3)))# would mean finding #32^3# first ... ouch!

#(color(blue)(root5 (32)))^3# would mean finding #root5(32)# first. That is #color(blue)(2)#

#(color(blue)(root5 (32)))^3 = color(blue)(2)^3 = 8#