# How do you find the square root of 74889?

Aug 21, 2017

The simplest form of the square root is $3 \sqrt{8321}$

We can find approximations such as:

$\sqrt{74889} \approx 273.659$

#### Explanation:

Given:

$74889$

Notice that the sum of the digits is divisible by $9$. That is:

$7 + 4 + 8 + 8 + 9 = 36 = 4 \cdot 9$

So $74889$ is divisible by $9$ ($= {3}^{2}$) too:

$74889 = 9 \cdot 8321$

$8321$ is less easy to factorise, but eventually you might find:

$8321 = 53 \cdot 157$

In fact, to check for square factors we only needed to look for prime factors up to $19$, since $20$ and $21$ are composite and:

${21}^{3} = 9261 > 8321$

Since there are no more square factors, the simplest form of the square root is given by:

$\sqrt{74889} = \sqrt{{3}^{2} \cdot 8321} = \sqrt{{3}^{2}} \cdot \sqrt{8321} = 3 \sqrt{8321}$

This is an irrational number, not expressible as a fraction, but we can find rational approximations:

Given:

$74889$

First split into pairs of digits from the right:

$7 \text{|"48"|"89}$

Note that:

${2}^{2} = 4 < 7 < 9 = {3}^{2}$

Hence:

$2 < \sqrt{7} < 3$

and:

$200 < \sqrt{74889} < 300$

For a better estimate, if we know a few more square roots we can include the next two digits and note that:

${27}^{2} = 729 < 748 < 784 = {28}^{2}$

Hence:

$270 < \sqrt{74889} < 280$

We can linearly interpolate between these limits to find:

$\sqrt{74889} \approx 270 + 10 \cdot \frac{74889 - 72900}{78400 - 72900} = 270 + 10 \cdot \frac{1989}{5500} \approx 273.6$

Let us choose $274$ as an approximation.

Given an approximation $a$ for the square root of a number $n$, a better approximation is given by:

$\frac{{a}^{2} + n}{2 a}$

So in our case, putting $n = 74889$ and $a = 274$ we find:

$\sqrt{74889} \approx \frac{{274}^{2} + 74889}{2 \cdot 274} = \frac{75076 + 74889}{548} = \frac{149965}{548} \approx 273.659$

If we want more accuracy, then repeat with this new approximation. Each iteration will roughly double the number of significant digits which are correct.

Aug 21, 2017

$\sqrt{74889} \approx 273.658$

#### Explanation:

As $74889$ is not a perfect square, to find the square root of $74889$, we should do a special long division, where we pair, the numbers in two, starting from decimal point in either direction. When we group them we do so starting from $89$, then $48$ and then $7$.

Here for $7$, the number whose square is just less than it, is $2$, whose square is $4$ and so we write $4$ below $7$. The difference is $3$ and now we bring down next two digits $48$. As a divisor we first write double of $2$ i.e. $4$ and then find a number $x$ so that $4 x$ (here $x$ stands for single digit in units place) multiplied by $x$ is just less than the number, here $348$. We find for $x = 7$, we have $47 \times 7 = 329$ and get the difference as $19$.

Next we bring down next two digits $89$ and we have $1989$. Recall we had brought as divisor $2 \times 2 = 4$, but this time we have $27$ double of which is $54$, so we make the divisor as $54 x$, selecting an $x$ so that $54 x$ multiplied with $x$, just goes in $1989$ and this is $3$, for which we get $543 \times 3 = 1629$ and have a remainder $360$.

Now as we still have a remainder of $360$, for more accuracy, we bring $00$ after decimal point. Also put decimal after $273$.

We continue in similar way by bringing down $00$, which makes it $36000$ and continue as shown till we have desired accuracy. You can see the details below, where we have gone up to three decimal places.

$\textcolor{w h i t e}{x X} 2 \textcolor{w h i t e}{\times} 7 \textcolor{w h i t e}{\times} 3 \textcolor{w h i t e}{\times} .6 \textcolor{w h i t e}{\times} 5 \textcolor{w h i t e}{\times} 8$
$\underline{2} | \overline{7} \textcolor{w h i t e}{.} \overline{48} \textcolor{w h i t e}{.} \overline{89} . \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00} \textcolor{w h i t e}{.} \overline{00}$
$\textcolor{w h i t e}{X X} \underline{4} \textcolor{w h i t e}{.} \downarrow$
$\textcolor{red}{4} 7 | 3 \textcolor{w h i t e}{X} 48$
$\textcolor{w h i t e}{\times x} \underline{3 \textcolor{w h i t e}{. .} 29}$
$\textcolor{w h i t e}{x} \textcolor{red}{54} 3 | \textcolor{w h i t e}{.} 19 \textcolor{w h i t e}{x} 89$
$\textcolor{w h i t e}{\times x . \times} \underline{16 \textcolor{w h i t e}{.} 29}$
$\textcolor{w h i t e}{x} \textcolor{red}{546} 6 | \textcolor{w h i t e}{.} 3 \textcolor{w h i t e}{.} 60 \textcolor{w h i t e}{.} 00$
$\textcolor{w h i t e}{\times \times \times} \underline{3 \textcolor{w h i t e}{.} 27 \textcolor{w h i t e}{.} 96}$
$\textcolor{w h i t e}{\times} \textcolor{red}{5472} 5 | \textcolor{w h i t e}{} 32 \textcolor{w h i t e}{.} 04 \textcolor{w h i t e}{.} 00$
color(white)(xxXxxxx)ul(27color(white)(.)36color(white)(.)25
$\textcolor{w h i t e}{\times} \textcolor{red}{54730} 8 | 4 \textcolor{w h i t e}{.} 67 \textcolor{w h i t e}{.} 75 \textcolor{w h i t e}{.} 00$
color(white)(xxxxxxxx)ul(4color(white)(.)37color(white)(.)84color(white)(.)64
$\textcolor{w h i t e}{\times \times \times \times x .} 29 \textcolor{w h i t e}{.} 90 \textcolor{w h i t e}{.} 36$

Hence $\sqrt{74889} \approx 273.658$

Aug 21, 2017

Here's another method for finding rational approximations...

#### Explanation:

For interest, here's another idea for finding rational approximations to $\sqrt{74889}$.

Start by noting that:

$\sqrt{{a}^{2} + b} = a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}}$

Given that $274$ is the closest integer approximation, with ${274}^{2} = 75076 = 74889 + 187$, we can write a generalised continued fraction for the square root, putting $a = 274$ and $b = - 187$ to get:

$\sqrt{74889} = 274 - \frac{187}{548 - \frac{187}{548 - \frac{187}{548 - \frac{187}{548 - \ldots}}}}$

This is related to $274 + \sqrt{74889}$ being one of the zeros of:

$\left(x - 274 - \sqrt{74889}\right) \left(x - 274 + \sqrt{74889}\right) = {x}^{2} - 548 x + 187$

Now consider a sequence defined recursively as follows:

$\left\{\begin{matrix}{a}_{0} = 0 \\ {a}_{1} = 1 \\ {a}_{n + 2} = 548 {a}_{n + 1} - 187 {a}_{n} \text{ for } n \ge 0\end{matrix}\right.$

The first few terms are:

$0 , 1 , 548 , 300117 , 164361640 , 90014056841 , 49296967522188$

Because of the way it is constructed, the ratio between pairs of successive terms tends to $274 + \sqrt{74889}$. In fact each successive ratio corresponds to truncating the continued fraction after one more term.

So we can use this sequence to get successively better approximations to $\sqrt{74889}$, such as:

$\sqrt{74889} \approx \frac{164361640}{300117} - 274 \approx 273.6585465$