lets think about range of x

0<x<1 and x>1

if x>1 then if x>y x^3>1

but if x<1 then even if x>y it may happen that x^3 < y

stmt 1:

x^1/2 > y

if X>1 then X^3 > 1 but if X<1 even if x^1/2 >y it does not imply x^3 >y INSUF

stmt 2 INSUFF by logic given above

taking two

x> y^2 and x> y still for fraction it may happen that x^3< y so INSUFF

choose E.

I noticed that the GMAT loves to test your knowledge of fractions with exponents and how they get smaller as they are raised to a power (or a square root of a fraction gets bigger, like in this example).

Remember to always check with negative numbers and fractions unless the problem explicitly states otherwise.

E for me as well.
take both the statements together
take x=.5 and y = .2 => x^3 < y
take x=4 and y = 1 => x^3 > y
Hence insufficient.

can I solve this problem by

1.

y<x^3 only when x<x^3

only when x>1

and there is infomation of x

not sufficient

we do the same for 2,

E is correct.

Can I do like this