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

#Algebra

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Comments

You could try expressing it as a continued fraction

let,

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

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.

Anirudh Sreekumar - 3 years, 8 months 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?

Alex Li - 3 years, 7 months ago

You may use differential calculus to find approximate value

Dev Verma - 3 years, 2 months ago

rt(71 x 3 x 4) = 2 x rt(71 x 3) = 2 x (8.4 x 1.7) = 2(14.28) = 28.56 (approx)

v a - 2 years, 7 months ago

For approximating using differential calculus, we use $f(a+h) ≈ f(a) + h f'(a)$ . This can help us approximate the value by taking $h$ a small number and $f(x) = \sqrt{x}$ . I noticed that $900 - 868 = 32$ . Hence, substituting $a = 900$ and $h = -32$ in the above formula, $\sqrt{900-32} ≈ \sqrt{900} - \frac{32}{2\sqrt{900}}$ , $\therefore \sqrt{868} ≈ 30 - \frac{32}{60}$ , ie. $\sqrt{868} ≈ 30 - \frac{8}{15} ≈ 29.46667$ . According to my calculator, $\sqrt{868} = 29.46184$ . So, the answers are about the same. You can take $h$ even closer to $0$ to get a more accurate approximation.

Sumant Chopde - 2 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$