Question

Describe how you would estimate the square root of a number that is not a perfect square with out using a calculator.

IF anyone could help me to have an understanding of what this question wants or means I'd appreciate it. Thanks

Answer 1

given a^2
guess x for sqrt a
a^2 = (x + y)^2
a^2 = x^2 + 2 x y + y^2
if my guess was perfect, y = 0
if just a little off, then y^2 is much less than 2 x y
so I approximate
2 x y = a^2 - x^2

for example if I want the square root of 10
a^2 = 10
guess x = 3 so x^2 = 9
2 (3) y = 10 - 9 = 1
y = 1/6
so my next guess at sqrt 10 is 3+1/6 or 3.167 = new x
well how far off is that?
3.167^2 = 10.03 , not bad. The next try using x = 3.167 will be about as close as your calulator could come:)
(my calculator says sqrt 10 = 3.16227766 )

Answer 2

Thanks I understand it much better now.