How do you estimate square roots?

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How do you estimate square roots?

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Answer 1 · 2015-02-01T12:14:12

It depends on what you can use, of course. Let's say you need to find the square root of a number $x$ $x$

A first, but very rough way, consists in simply finding two numbers, $n$ $n$ and $m$ $m$ , such that $n^{2} < x < m^{2}$ $n^2 < x < m^2$ . If you have this relation, you can surely affirm that $\sqrt{x}$ $\sqrt{x}$ is some number between $n$ $n$ and $m$ $m$ . For example, if we needed to estimate $\sqrt{40}$ $\sqrt{40}$ , we could say that it surely is a number between $6$ $6$ and $7$ $7$ .

Of course, this method can be used also with rational numbers. For example, with a bit of calculations you can find out that $\sqrt{40}$ $\sqrt{40}$ actually lies between $6.3$ $6.3$ and $6.4$ $6.4$ , and so on.

Another way could be factoring $x$ $x$ with primes, and simplify squared factors, if any appear. This could leave only smaller roots to calculate: consider for example $\sqrt{18}$ $\sqrt{18}$ . You can write $18$ $18$ as $2 \cdot 3^{2}$ $2\cdot 3^2$ , and so $\sqrt{18} = \sqrt{2 \cdot 3^{2}} = \sqrt{2} \cdot \sqrt{3^{2}} = 3 \sqrt{2}$ $\sqrt{18}=\sqrt{2\cdot 3^2}=\sqrt{2}\cdot\sqrt{3^2}=3\sqrt{2}$

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How do you estimate square roots?

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