$$12\times 12 = 144$$ If there's a question that asks me to find square roots manually, how will I find that? I know a way that's really long and boring. I actually know about the "Tree" method. Is there any other interesting and effective way to find square root?
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Are you familiar with the “tree” method for square roots? – Taylor Rendon May 25 '20 at 14:35
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1Does this answer your question? Is there any simple method to calculate $\sqrt x$ without using logarithm – don-joe May 25 '20 at 14:38
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1no,what is the 'tree' method? – Himel May 25 '20 at 14:38
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Another good way to think about the square root function: the square root function can be thought of as a map from the area of a square to its side length. We all know how to find the area of a square right? In that light, you could always think “if the square has area 144... which number must I take the square of (I.e. $a^2$ where $a$ is the side length of the square) in order to get the area? – Taylor Rendon May 25 '20 at 14:40
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Also, here’s a great video on the “factor tree method” for square roots : https://m.youtube.com/watch?v=tBSc7UCzDAU – Taylor Rendon May 25 '20 at 14:42
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2If the root is $x$, then we should have $x=144/x$. Make a guess, say $x=10$. Compare that to $144/10 = 14.4$. Take the average of these two: $\frac12(10+14.4) = 12.2$. That's your new guess (and it's already close to the answer; you could try $12$ at this point). Repeat. – Théophile May 25 '20 at 14:46
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yes, I know TREE method. I did not know its name. – Himel May 25 '20 at 14:48
3 Answers
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To get the square root of $x$ without calculator use the following steps:
1. Make a rough estimate or guess.
2. Divide $x$ by your estimate.
3. Take the average of the answer from 2. and your estimate ....this will become your new estimate.
(Repeat steps 2 and 3 until you reach the accuracy you want.)
In equation form...where:
$x$ is the number you want to find the square root of, $a_1$ is your initial estimate, $a_2$ is your first calculated improved estimate, $a_3$ is your next calculated improved estimate, ...
\begin{equation} a_1 = initial ~estimate~ or~ guess \\ a_2 = \frac{(x/a_1 + a_1) }{2}\\ a_3 = \frac{(x/a_2 + a_2)}{2} \\ a_4 = \frac{(x/a_3 + a_3)}{2} \\ ... \end{equation}
Further,
Suppose you want to calculate the root of a positive number $M$.
Assume $x=\sqrt M$,then
$f(x)=x^2-M=0$.
Now use the Newton-Raphson Method.Newton-Raphson method For this equation is given by \begin{equation} x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} \end{equation}
Unknown
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To calculate square root, we need to start making groups, taking $2$ digits at a time, starting from right side. Here, $44$ is one group. And $1$ is another group. Now, starting from the leftmost group i.e. $1$ here. We need to think of a square number less than or equal to the left most group. Here, that square number is $1$. Subtracting $1$ from $1$, we get $0$. Now we need to focus on $44$. For that, let's double our quotient here i.e. $1$. We get $2$. On its right side, we need to place a digit so that the number thus formed when multiplied by that added digit gives us a number less than or equal to our remainder, which is $44$ here. So, we add $2$ to the right of $2$, we get $22$. Multiplying it with $2$, we get $44$. Subtracting it from $44$, we get zero. It means our square root is $12$. I'll add another example to this answer to make the concept clearer.
aarbee
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1that is clear to me. Because I know this method. do you have any other process? – Himel May 25 '20 at 15:03
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Here is what I would do. The smallest square is $4$. Does $4$ divide evenly into $144$? Yes! $\frac{144}{4} = 36.$ $4$ divides into that again- $\frac{36}{4} = 9.$ And $9 = 3\cdot3.$ So $144 = 3\cdot3\cdot4\cdot4 = 3^2\cdot4^2$, so the square root of $144$ is $3\cdot4 = 12.$
user247327
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