Simplify the following expression:
(x+y)(x−y)(x
2
+y
2
)(x
4
+y
4
)(x
8
+y
8
)(x
1
6+y
1
6)

We can simplify the expression step-by-step using the algebraic identity:
(a−b)(a+b)=a
2
−b
2

Step 1: Combine (x−y)(x+y):

(x−y)(x+y)=x
2
−y
2

Now the expression becomes:

(x
2
−y
2
)(x
2
+y
2
)(x
4
+y
4
)(x
8
+y
8
)(x
16
+y
16
)
Step 2: Combine (x
2
−y
2
)(x
2
+y
2
):

(x
2
−y
2
)(x
2
+y
2
)=x
4
−y
4

Step 3: Combine (x
4
−y
4
)(x
4
+y
4
):

(x
4
−y
4
)(x
4
+y
4
)=x
8
−y
8

Step 4: Combine (x
8
−y
8
)(x
8
+y
8
):

(x
8
−y
8
)(x
8
+y
8
)=x
16
−y
16

Step 5: Combine (x
16
−y
16
)(x
16
+y
16
):

(x
16
−y
16
)(x
16
+y
16
)=x
32
−y
32