If
y
x
​
+
x
y
​
=1, and x,y

=0, then find the value of x
6
+y
6
+2x
3
y
3
.

Given equation:
y
x
​
+
x
y
​
=1
Taking the LCM on the left-hand side:

xy
x
2
+y
2

​
=1⟹x
2
+y
2
=xy⟹x
2
−xy+y
2
=0
We know the algebraic identity for the sum of cubes:

x
3
+y
3
=(x+y)(x
2
−xy+y
2
)
Substituting x
2
−xy+y
2
=0 into the identity:

x
3
+y
3
=(x+y)(0)=0
Now we need to find the value of x
6
+y
6
+2x
3
y
3
.
Notice that this expression can be written as a perfect square:

x
6
+y
6
+2x
3
y
3
=(x
3
)
2
+(y
3
)
2
+2(x
3
)(y
3
)=(x
3
+y
3
)
2

Since x
3
+y
3
=0:

(x
3
+y
3
)
2
=0
2
=0