# How do you find the square root of 52?

Sep 12, 2015

$\sqrt{52} = 2 \sqrt{13} \approx 7.21110$

#### Explanation:

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$, so:

$\sqrt{52} = \sqrt{{2}^{2} \cdot 13} = \sqrt{{2}^{2}} \sqrt{13} = 2 \sqrt{13}$

If you want to calculate an approximation by hand, you could follow the advice I gave for $\sqrt{28}$ in http://socratic.org/questions/how-do-you-find-the-square-root-28

Using the method described there:

Let $n = 52$, ${p}_{0} = 7$, ${q}_{0} = 1$

Then:

${p}_{1} = {7}^{2} + 52 \cdot {1}^{2} = 49 + 52 = 101$
${q}_{1} = 2 \cdot 7 \cdot 1 = 14$

${p}_{2} = {101}^{2} + 52 \cdot {14}^{2} = 10201 + 10192 = 20393$
${q}_{2} = 2 \cdot 101 \cdot 14 = 2828$

Stopping at this point, we get a result accurate to $5$ decimal places:

$\sqrt{52} \approx \frac{20393}{2828} \approx 7.21110$