# How do you simplify sqrt(442)?

Nov 7, 2015

$\sqrt{442}$ cannot be simplified, but it has a simple continued fraction expansion:

sqrt(442) = [21;bar(42)] = 21 + 1/(42+1/(42+1/(42+...)))

#### Explanation:

$442 = 2 \cdot 13 \cdot 17$ has no square factors, so we cannot simplify the square root.

It is an irrational number, so it cannot be expressed in the form $\frac{p}{q}$ for any integers $p$ and $q$. Neither will its decimal expansion terminate or repeat.

However, it is of the form ${n}^{2} + 1$, since $442 = 441 + 1 = {21}^{2} + 1$.

As a result, the continued fraction for its square root takes a particularly simple form:

sqrt(442) = [21;bar(42)] = 21 + 1/(42+1/(42+1/(42+...)))

In general, any positive integer of the form ${n}^{2} + 1$ has square root:

sqrt(n^2+1) = [n;bar(2n)]