# How do you find the square root of 12025?

Feb 5, 2017

$\sqrt{12025} = 5 \sqrt{481} \approx 109.658561$

#### Explanation:

Note that:

$12025 = {5}^{2} \cdot 481 = {5}^{2} \cdot 13 \cdot 37$

So:

$\sqrt{12025} = \sqrt{{5}^{2} \cdot 481} = 5 \sqrt{481}$

We can find rational approximations for $\sqrt{481}$ using a form of the Babylonian method:

Given a rational approximation $\frac{p}{q}$ to $\sqrt{n}$, we can find a better approximation by calculating:

$\frac{{p}^{2} + n {q}^{2}}{2 p q}$

In our example, $481$ is quite close to $484 = {22}^{2}$, so use $\frac{22}{1}$ as our first approximation.

The next approximation would be:

$\frac{{22}^{2} + 481 \cdot {1}^{2}}{2 \cdot 22 \cdot 1} = \frac{484 + 481}{44} = \frac{965}{44}$

For more accuracy, iterate again:

$\frac{{965}^{2} + 481 \cdot {44}^{2}}{2 \cdot 965 \cdot 44} = \frac{931225 + 931216}{84920} = \frac{1862441}{84920} \approx 21.9317122$

So:

$\sqrt{12025} \approx 5 \cdot 21.9317122 = 109.658561$