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# Trust me you can

Can you show how you get the value of $$\large\sqrt{868}$$ without a calculator?

Note by Sakib Nazmus
3 months, 2 weeks ago

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You could try expressing it as a continued fraction

let,

\begin{align} x&= \sqrt{868}\\ \lfloor{x\rfloor}&=29\\ x^2-29^2&=27\\ (x+29)(x-29)&=27\\ x&=29+\dfrac{27}{(x+29)}\end{align}

Substituting for $$x$$ repeatedly we get the continued fraction of $$x$$,i.e.

$$x=29+\dfrac{27}{58+\dfrac{27}{58+\dfrac{27}{58+\dfrac{27}{58+_\ddots}}}}$$

You could truncate the expression at about 2 or 3 levels and end up with a fairly decent approximation.

- 3 months, 2 weeks ago

I'm really confused... how can you simplify it more? Square roots are irrational, right? Do you mean the actual value or just an approximation?

- 2 months, 2 weeks ago