Question

describe how you would estimate the square root of a number that is not a perfect square without using a calculator

Answer 1

well know the square numbers

\[1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36....\]

so if I asked \[\sqrt{30}\]

I know 30 is between 5^2 and 6^2 a little closer to 6^2 so I'd approximate it at 5.54 or something close to that.

hope it makes sense

Answer 2

That is an algebraic way and I like it very much.
You can also look at some calculus ways if you know any calculus.

Answer 3

Do you know any calculus?

Answer 4

\[\sqrt{x}=\sqrt{30} \text{ when } x=30 \\ \text{ so I will use } f(x)=\sqrt{x} \text{ as my function } \\ \text{ I want to find the tangent line to } f \text{ at a value that is very close to 30 like 25 } \\ f(25)=\sqrt{25}=5 \\ \text{ if } f(x)=\sqrt{x} \text{ then } f'(x)=\frac{1}{2 \sqrt{x}} \\ \text{ and } f'(25)=\frac{1}{2 \sqrt{25}}=\frac{1}{2(5)}=\frac{1}{10} \\ \text{ so the tangent line at } x=25 \text{ is } \\ y-5=\frac{1}{10}(x-25) \\ y-5=\frac{x-25}{10} \\ y=\frac{x-25}{10}+5 \\ \text{ so } f(x)=\sqrt{x} \approx \frac{x-25}{10}+5 \text{ for values of } x \text{ near } x=25 \\ \text{ we wanted to approximate } f \text{ at } x=30 \\ \sqrt{30} \approx \frac{30-25}{10}+5=\frac{5}{10}+5=\frac{1}{2}+5=5.5\]

Answer 5

I chose 25 because I knew the square root of 25

Answer 6

and it was also close to 30

Answer 7

oh Thank both of you that helped a lot