SOLUTION: The question is as followed: Without the use of a calculator explain how in the olden days they used to evaluate the following. Square Root of Two (2) √2 Square Root

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Question 205460: The question is as followed:
Without the use of a calculator explain how in the olden days they used to evaluate the following.
Square Root of Two (2) √2
Square Root of Seven (7) √7
" " " Thirteen (13) √13
" " " One Hundred and sixty nine (169) √169
I'm unsure where to ever start with this.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll show you how to find the square root of 2:

Step 1) Divide 2 by 1 to get 2. Since 1%3C%3E2, this means that the square root of 2 is NOT 1.


Step 2) Average the previous result 2 with the first number 1 to get: %281%2B2%29%2F2=3%2F2=1.5


Step 3) Divide 2 by 1.5 to get 1.33333... Since 1.5%3C%3E1.3333, this means that the square root of 2 is NOT 1.5


Note: you can use long division by writing it out by hand.


Step 4) Average the quotient 1.3333... with 1.5 to get %281.3333%2B1.5%29%2F2=1.41665


Step 5) Divide 2 by 1.41665 to get 1.41178 (take note how the numbers are getting closer). Since 1.41665%3C%3E1.41178, this means that the square root of 2 is NOT 1.41665 (or 1.41178). But we are getting closer


Step 6) Average 1.41665 and 1.41178 to get %281.41665%2B1.41178%29%2F2=1.4142


Step 7) Divide 2 by 1.4142 to get 1.414227. If we round the this quotient to the nearest ten-thousandth, we get 1.4142. Since this 2%2F1.4142=1.4142 (to four places), this means that 1.4142%2A1.4142=2 (approximately). In other words, %281.4142%29%5E2=2


So to four decimal places, the square root of 2 is roughly 1.4142


So



Follow the same logic to find the square root of 7, 13, and 169.