What is the square root of 8 to the 3rd power?

I don't know whether you mean ##sqrt(8^3)## or ##sqrt8^3##.

The good news is: it doesn't matter, they are equal.

There is only only number here, but there are several ways to write it and several ways to get to the "simplist form" of ##16sqrt2##

##sqrt(8^3)=sqrt(8*8*8)=sqrt(8^2*8)=sqrt8^2*sqrt8=8sqrt8##

Now, ##sqrt8=sqrt(4*2)=sqrt4*sqrt2=2sqrt2## so we can continue:

##sqrt(8^3)=8sqrt8=8sqrt(4*2)=8sqrt4*sqrt2=8*(2*sqrt2)=(8*2)sqrt2=16sqrt2##

OR ##sqrt(8^3)=sqrt((2^3)^3)=sqrt(2^(3*3) )= sqrt(2^9)=sqrt(2^8*2)##

##=sqrt(2^8)sqrt2=sqrt((2^4)^2)sqrt2=2^4sqrt2=16sqrt2##

OR ##sqrt8^3=sqrt8^2*sqrt8=8sqrt8=8sqrt(4*2)=8sqrt4*sqrt2=8*2*sqrt2=16sqrt2##