Question 205460
I'll show you how to find the square root of 2:


Step 1) Divide 2 by 1 to get 2. Since {{{1<>2}}}, 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: {{{(1+2)/2=3/2=1.5}}}



Step 3) Divide 2 by 1.5 to get 1.33333...  Since {{{1.5<>1.3333}}}, 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 {{{(1.3333+1.5)/2=1.41665}}}



Step 5) Divide 2 by 1.41665 to get 1.41178 (take note how the numbers are getting closer). Since {{{1.41665<>1.41178}}}, 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 {{{(1.41665+1.41178)/2=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 {{{2/1.4142=1.4142}}} (to four places), this means that {{{1.4142*1.4142=2}}} (approximately). In other words, {{{(1.4142)^2=2}}}



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



So *[Tex \LARGE \sqrt{2} \approx 1.4142]




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