Explanation:

Whenever you meet x³+y³=9 and x+y=3 together straightway go for the expression x³+y³=(x+y)×(x²−xy+y²) which results in,
9=3×((x+y)²−3xy)=3×(9−3xy)=27−9xy
or, xy=2.
Now easiest way to get x4+y4 is to multiply the first two given expressions together, giving,
x⁴+y⁴+xy(x²+y²)=27. So,
x⁴+y⁴=27−2×((x+y)²−2xy)=27−2×(9−4)=17