1. Find Square root of 4096 using Babylonian method, with Initial guess x0 = 60

Solution:
Babylonian method for finding square roots by hand
`sqrt(4096)=?`

`x_(i+1)=(x_i+s/x_i)/2`

`x_0=60`


`1^(st)` iteration :

`x_1=(60+4096/60)/2`

`=(60+68.2667)/2`

`=(128.2667)/2`

`=64.1333`


`2^(nd)` iteration :

`x_2=(64.1333+4096/64.1333)/2`

`=(64.1333+63.8669)/2`

`=(128.0003)/2`

`=64.0001`


`3^(rd)` iteration :

`x_3=(64.0001+4096/64.0001)/2`

`=(64.0001+63.9999)/2`

`=(128)/2`

`=64`


`4^(th)` iteration :

`x_4=(64+4096/64)/2`

`=(64+64)/2`

`=(128)/2`

`=64`

`:. sqrt(4096)=64`

Babylonian Method Table

Iteration	`x`	`4096/x`	Average
1	60	68.2667	64.1333
2	64.1333	63.8669	64.0001
3	64.0001	63.9999	64
4	64	64	64

2. Find Square root of 70 using Babylonian method, with Initial guess x0 = 8

Solution:
Babylonian method for finding square roots by hand
`sqrt(70)=?`

`x_(i+1)=(x_i+s/x_i)/2`

`x_0=8`


`1^(st)` iteration :

`x_1=(8+70/8)/2`

`=(8+8.75)/2`

`=(16.75)/2`

`=8.375`


`2^(nd)` iteration :

`x_2=(8.375+70/8.375)/2`

`=(8.375+8.3582)/2`

`=(16.7332)/2`

`=8.3666`


`3^(rd)` iteration :

`x_3=(8.3666+70/8.3666)/2`

`=(8.3666+8.3666)/2`

`=(16.7332)/2`

`=8.3666`

`:. sqrt(70)=8.3666`

Babylonian Method Table

Iteration	`x`	`70/x`	Average
1	8	8.75	8.375
2	8.375	8.3582	8.3666
3	8.3666	8.3666	8.3666

3. Find Square root of 256 using Babylonian method, with Initial guess x0 = 15

Solution:
Babylonian method for finding square roots by hand
`sqrt(256)=?`

`x_(i+1)=(x_i+s/x_i)/2`

`x_0=15`


`1^(st)` iteration :

`x_1=(15+256/15)/2`

`=(15+17.0667)/2`

`=(32.0667)/2`

`=16.0333`


`2^(nd)` iteration :

`x_2=(16.0333+256/16.0333)/2`

`=(16.0333+15.9667)/2`

`=(32.0001)/2`

`=16`


`3^(rd)` iteration :

`x_3=(16+256/16)/2`

`=(16+16)/2`

`=(32)/2`

`=16`

`:. sqrt(256)=16`

Babylonian Method Table

Iteration	`x`	`256/x`	Average
1	15	17.0667	16.0333
2	16.0333	15.9667	16
3	16	16	16

4. Find Square root of 101 using Babylonian method, with Initial guess x0 = 10

Solution:
Babylonian method for finding square roots by hand
`sqrt(101)=?`

`x_(i+1)=(x_i+s/x_i)/2`

`x_0=10`


`1^(st)` iteration :

`x_1=(10+101/10)/2`

`=(10+10.1)/2`

`=(20.1)/2`

`=10.05`


`2^(nd)` iteration :

`x_2=(10.05+101/10.05)/2`

`=(10.05+10.0498)/2`

`=(20.0998)/2`

`=10.0499`


`3^(rd)` iteration :

`x_3=(10.0499+101/10.0499)/2`

`=(10.0499+10.0499)/2`

`=(20.0998)/2`

`=10.0499`

`:. sqrt(101)=10.0499`

Babylonian Method Table

Iteration	`x`	`101/x`	Average
1	10	10.1	10.05
2	10.05	10.0498	10.0499
3	10.0499	10.0499	10.0499

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