If x
4
+y
4
+x
2
y
2
=21 and x
2
+y
2
−xy=7, then what is the value of
x
2

1
​
+
y
2

1
​
?

We know the algebraic identity:
x
4
+y
4
+x
2
y
2
=(x
2
+y
2
+xy)(x
2
+y
2
−xy)
Given x
4
+y
4
+x
2
y
2
=21 and x
2
+y
2
−xy=7:

21=(x
2
+y
2
+xy)×7⟹x
2
+y
2
+xy=3— (Equation 1)
And we have:

x
2
+y
2
−xy=7— (Equation 2)
Adding Equation 1 and Equation 2:

2(x
2
+y
2
)=10⟹x
2
+y
2
=5
Subtracting Equation 2 from Equation 1:

2xy=−4⟹xy=−2⟹(xy)
2
=x
2
y
2
=4
We need to find the value of:

x
2

1
​
+
y
2

1
​
=
x
2
y
2

x
2
+y
2

​
=
4
5
​