If x+
x
1
​
=1−
2

​
, then find the value of x
3
+
x
3

1
​
.

Let x+
x
1
​
=v=1−
2

​
.
We know the standard identity:

x
3
+
x
3

1
​
=(x+
x
1
​
)
3
−3(x+
x
1
​
)=v
3
−3v
Let's compute v
3
−3v:

v
3
−3v=v(v
2
−3)
First, find v
2
:

v
2
=(1−
2

​
)
2
=1+2−2
2

​
=3−2
2

​

Now substitute v
2
into the expression (v
2
−3):

v
2
−3=(3−2
2

​
)−3=−2
2

​

Now, multiply by v:

v(v
2
−3)=(1−
2

​
)(−2
2

​
)=−2
2

​
+2(2)=4−2
2

​