The powers of 2 form a sequence:
20  =   1
21  =   2
22  =   4
23  =   8
24  =  16
25  =  32
26  =  64
27  = 128

The sums of the rows in Pascal's triangle are equal to the powers of 2:

 1 1 Row 0 1 1 2 Row 1 1 2 1 4 Row 2 1 3 3 1 8 Row 3 1 4 6 4 1 16 Row 4 1 5 10 10 5 1 32 Row 5 1 6 15 20 15 6 1 64 Row 6 1 7 21 35 35 21 7 1 128 Row 7 1 8 28 56 70 56 28 8 1 Row 8

The powers of 11 also form a sequence:

110  =   1
111  =   11
112  =   121
113  =   1331
114  =   14641
115  =   161051
116  =   1771561
117  =   19487171

The powers of 11 can be extracted from Pascal's triangle by reading across the rows and interpreting the digits as a place value system. Starting in row 5 the pattern becomes harder to see, because a two-digit number like the number 10 can not occupy a single place. You can think of row 5 in this way:

1(105) + 5(104) + 10(103) + 10(102) + 5(101) + 1(100)

= 100000 + 50000 + 10000 + 1000 + 50 + 1

= 161051

 1 1 Row 0 1 1 11 Row 1 1 2 1 121 Row 2 1 3 3 1 1331 Row 3 1 4 6 4 1 14641 Row 4 1 5 10 10 5 1 161051 Row 5 1 6 15 20 15 6 1 1771561 Row 6 1 7 21 35 35 21 7 1 19487171 Row 7 1 8 28 56 70 56 28 8 1 Row 8

Let's look closely at Row 6:

 1 6 15 20 15 6 1 1771561 Row 6 1 (6+1) (5+2) (0+1) 5 6 1 1 7 7 1 5 6 1 1771561

And here's another way to think about the powers of 11:

11 = (10+1)
Let x = 10

 11^2 = (x+1)^2 = 1x^2 + 2x + 1 = 100 + 20 + 1 = 121 11^3 = (x+1)^3 = 1x^3 + 3x^2 + 3x + 1 = 1000 + 300 + 30 + 1 = 1331 11^4 = (x+1)^4 = 1x^4 + 4x^3 + 6x^2 + 4x + 1 = ...
References
1. Ask Dr. Math Archives: Exponents
2. Eric Weisstein's World of Mathematics: Power