a) Compare the first four powers of 11 with entries in Pascal's triangle. Describe any pattern you notice. b) Explain how you could express row 5 as a power of 11 by regrouping the entries. c)...

a) Compare the first four powers of 11 with entries in Pascal's triangle. Describe any pattern you notice.

b) Explain how you could express row 5 as a power of 11 by regrouping the entries.

c) Demonstrate how to express rows 6 and 7 as powers of 11 using the regrouping method from part b). Describe your method clearly.

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The first few rows of Pascal's triangle are:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

`11^0=1`

`11^1=11`

`11^2=121`

`11^3=1331`

`11^4=14641`

Now `11^5=161051`

The corresponding row is 1 5 10 10 5 1 -- the rightmost digits are (reading from right to left) 1 and 5; since 10 has 2 digits we place a zero and carry a 1; the next "digit" is 11 so place a 1 and carry a 1; the next digit is 4+ the carry so 5 and finally 1.

`11^6=1771561`

Row 6 is 1 6 15 20 15 6 1 -- so reading from right to left: place a 1, then a 6, then a 5 (carry a 1), then a 1 (carry a 2), then a 7 (carry a 1),then a 7 and finally a 1.

Row 7 is 1 7 21 35 35 21 7 1

To find `11^7` : reading from right to left 1,7,1 (carry a 2); the next entry is 35+2=37 so place a 7 and carry a 3; the next entry is 35+3=38 so place an 8 and carry a 3; the next entry is 21+3=24 so place a 4 and carry a 2; the next entry is 7+2=9, and finally a 1 yielding:

`11^7=19487171`

Sources:

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