If (x
2
+
x
2

1
​
)=18 and x>1, what is the value of (x
3
−
x
3

1
​
)?

Given that:
x
2
+
x
2

1
​
=18
Subtracting 2 from both sides to form a perfect square:

x
2
+
x
2

1
​
−2=18−2
(x−
x
1
​
)
2
=16
Taking the square root on both sides (since x>1, x−
x
1
​
is positive):

x−
x
1
​
=4
Now, we use the standard algebraic identity for the difference of cubes:

x
3
−
x
3

1
​
=(x−
x
1
​
)
3
+3(x−
x
1
​
)
Substitute x−
x
1
​
=4:

x
3
−
x
3

1
​
=(4)
3
+3(4)
x
3
−
x
3

1
​
=64+12=76