If x
3
=184+y
3
and x=4+y, then the value of (x+y) is (given that x>0 and y>0):

Given equations:
x
3
−y
3
=184

x−y=4

We know the algebraic identity:

x
3
−y
3
=(x−y)(x
2
+xy+y
2
)
184=4(x
2
+xy+y
2
)
x
2
+xy+y
2
=46
We can express x
2
+xy+y
2
as:

x
2
+xy+y
2
=(x−y)
2
+3xy
46=(4)
2
+3xy
46=16+3xy
3xy=30⟹xy=10
Now we need to find the value of (x+y). We use the identity:

(x+y)
2
=(x−y)
2
+4xy
(x+y)
2
=4
2
+4(10)
(x+y)
2
=16+40=56
x+y=±
56

​
=±2
14

​

Since it is given that x>0 and y>0, their sum (x+y) must be positive.

x+y=2
14

​