# How do you find the square root of 16562?

Jun 8, 2016

$\sqrt{16562} \approx 128.699$

#### Explanation:

The ancient greeks used to compute square roots by sucessive approximations.

Given a number $N$ for which they need to compute the square root and given an initial approximation ${q}_{0}$ they proceed as follows:

${\left({q}_{0} + \delta {q}_{0}\right)}^{2} = N$ or
${q}_{0} + 2 {q}_{0} \delta q + {\left(\delta {q}_{0}\right)}^{2} = N$

So they were searching for an approximation variation $\delta {q}_{0}$ with the purpose of correct the initial guess ${q}_{0}$. They supposed also that $\delta {q}_{0}$ being small, much smaller would be ${\left(\delta {q}_{0}\right)}^{2}$ so they used the approximation

${q}_{0} + 2 {q}_{0} \delta {q}_{0} \approx N$

solving for $\delta {q}_{0}$ they got

$\delta {q}_{0} = \frac{\left(\frac{N}{q} _ 0\right) - {q}_{0}}{2}$

once corrected ${q}_{0}$ they got ${q}_{1} = {q}_{0} + \delta {q}_{0}$

$\delta {q}_{1} = \frac{\left(\frac{N}{q} _ 1\right) - {q}_{1}}{2}$ etc.

Let us apply that process for calculation of square root of

$N = 16562$

our initial guess will be ${q}_{0} = 400$

$\delta {q}_{0} = \frac{\left(\frac{16562}{400}\right) - 400}{2} = - 179.298$

so ${q}_{1} = 400 - 179.298$ and calculating $\delta {q}_{1}$

$\delta {q}_{1} = - 72.83015$ so ${q}_{2} = 400 - 179.298 - 72.83015$

calculating $\delta {q}_{2}$

$\delta {q}_{2} = - 17.93516$ so ${q}_{3} = 400 - 179.298 - 72.83015 - 17.93516$

In the third iteration we get

${\delta}_{3} = - 1.23779$ obtaining a result of

$\sqrt{16562} \approx 128.699$ which is a satisfactory result.

Jun 8, 2016

$\sqrt{16562} = 91 \sqrt{2} = 128.6922$

#### Explanation:

To find square root of $16562$, we should first factorize it.

From divisibility rules, it is apparent that it is divisible by $2$ and dividing by $2$, we get $8281$.

$8281$ is clearly not divisible by $3$ and $5$, but is divisible by $7$. Dividing by $7$, we get $1183$, which is again divisible by $7$and dividing it by $7$ we get $169$, which is $13 \times 13$.

Hence, $16562 = 2 \times 7 \times 7 \times 13 \times 13$ and hence

$\sqrt{16562} = \sqrt{2 \times \overline{7 \times 7} \times \overline{13 \times 13}}$

= $7 \times 13 \times \sqrt{2} = 91 \sqrt{2} = 91 \times 1.4142 = 128.6922$

Jun 9, 2016

$128.69$ to 2 decimal places

#### Explanation:

If you are not sure of the numbers build a factor tree to find the squared prime numbers.

Thus $\sqrt{16562} = \sqrt{2 \times {7}^{2} \times {13}^{2}}$

$= 7 \times 13 \times \sqrt{2} = 91 \sqrt{2} \approx 128.69$ to 2 decimal places