Question #eee37

We're going to start off with the 'numerator is the power,' a rule by which we can convert radicals to fractional exponents. The number by the radical is the denominator of the fractional exponent, and the power to which the radical's contents is raised is the numerator of the fractional exponent.

#rootcolor(red)5 4=4^(1/color(red)5)#

#rootcolor(red)3 4=4^(1/color(red)3)#

#rootcolor(red)2 4=4^(1/color(red)2)#

Note that we do not usually write the #2# on a square root sign because it is implied.

Anyway, we have three numbers multiplied together.
The bases (#4#) are the same, so we keep the base and add the exponents.

#4^(1/5)*4^(1/3)*4^(1/2)=4^(1/5+1/3+1/2)#

#4^(1/5+1/3+1/2)=4^(6/30+10/30+15/30)=4^(31/30)=4^(30/30)*4^(1/30)#

Since we have #4^(31/30)#, we can split it up:

#4^(31/30)=4^(30/30)*4^(1/30)#

And simplify the exponent:

#4^1*4^(1/30)=4*4^(1/30)#

Again using the 'numerator is the power,' we can turn #4^(1/30)# into a radical statement. #1#, the numerator will be the power of #4#, and #30#, the denominator, will be the number on the radical.

#4root30 4#