How to carry out the theory of exponents with roots...??

In general.......  ⁿ √[ a^m ]  =  a^(m/n) 

And    a^(m/n) * a^(r/s)  =  a ^{(m/n + r/s)}

So...for instance.....suppose we had

⁵ √ [4³ ]  *   ³ √ [ 4²]  ......we can write this in exponential form as

4 ^3/5  * 4 ^2/3  =     4^{(3/5 + 2/3)}   = 4 ^{( [ 9 + 10] / 15 )}  =  4 ^(19/15)

P.S. -  Note that ......   √a  in exponential form is understood to be  a^(1/2)