# How do you calculate the square root of 27.3529 without using a calculator?

Jul 15, 2015

Find $\sqrt{273529} = 523$ then divide by $100$ to get:

$\sqrt{27.3529} = 5.23$

#### Explanation:

$\sqrt{270000} = \sqrt{3 \cdot {3}^{2} \cdot {100}^{2}} = 300 \cdot \sqrt{3}$

Hopefully you know that $\sqrt{3} \cong 1.732$, so

$\sqrt{270000} = 300 \cdot \sqrt{3} \cong 300 \cdot 1.732 = 519.6$

Let's use $520$ as our first approximation for $\sqrt{273529}$

Let ${a}_{0} = 520$, $n = 273529$

Iterate using Newton Raphson method...

${a}_{i + 1} = {a}_{i} + \frac{n - {a}_{i}^{2}}{2 {a}_{i}}$

${a}_{1} = {a}_{0} + \frac{n - {a}_{0}^{2}}{2 {a}_{0}}$

$= 520 + \frac{273529 - {520}^{2}}{2 \cdot 520}$

$= 520 + \frac{273529 - 270400}{1040}$

$= 520 + \frac{3129}{1040}$

$\cong 520 + 3.00865$

$\cong 523$

Hmmm, that looks suspicious. Try $523 \cdot 523 = 273529$ - yes.