What is the Square root of 21?

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What is the Square root of 21?

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Answer 1 · 2015-10-15T19:29:37

$21 = 3 \cdot 7$ $21 = 3*7$ has no square factors, so is not possible to simplify $\sqrt{21}$ $sqrt(21)$

$\sqrt{21} \approx 4.583$ $sqrt(21) ~~ 4.583$ is an irrational number whose square is $21$ $21$

Explanation:

$\sqrt{21}$ $sqrt(21)$ is not a rational number, so it cannot be expressed as $\frac{p}{q}$ $p/q$ for some integers $p, q$ $p, q$ and its decimal expansion does not repeat.

$\sqrt{21} \approx 4.58257569495584000658$ $sqrt(21) ~~ 4.58257569495584000658$

It is expressible as a repeating continued fraction:

$sqrt(21) = [4;bar(1,1,2,1,1,8)] = 4 + 1/(1+1/(1+1/(2+...)))$

To see how to calculate this see http://socratic.org/questions/given-an-integer-n-is-there-an-efficient-way-to-find-integers-p-q-such-that-abs-#176764

We can get a good approximation for $sqrt(21)$ by truncating the continued fraction.

$sqrt(21) ~~ [4;1,1,2,1,1] = 4+1/(1+1/(1+1/(2+1/(1+1/1)))) = 55/12 = 4.58dot(3)$

This is a good approximation because $55^2 = 21*12^2 + 1$

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What is the Square root of 21?

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