Restriction on variables in these 3 square root of 3y^3 square root of 6x^2
(a) Given the expression `3sqrt(3y^3)` :
In the real number system, an even root is defined for nonnegative numbers only. Thus in the reals, `3sqrt(3y^3)` is defined only if `3y^3>=0` .
`3y^3>=0 ==> y>=0`
So the restriction on the variable (the domain of the expression) is `y>=0` .
*** If you meant `root(3)(3y^3)` then there is no restriction in the real number system. We can always take the cube root (or any odd indexed root) of any number in the reals. ***
(b) Given `sqrt(6x^2)` : again in the real numbers the expression in the radicand must be nonnegative.
`6x^2>=0 ==> x>=0`
Then the restriction in the reals is `x>=0` .
*** In the complex numbers (a number system with the imaginary unit) there is no restriction. ***