What is the process for doing the square root of a number? Like how do you do it? not on a calculator.?

Question

Like order of operations you use PEMDAS well what is the way you do square root/ hwo do you do it?

also this is only 8th grade math so don't explain something I haven't earned yet

1st I kno what PEMDAS is I was using it as an example for the question and 2 I don't understand that second one where does 64 and 5 come in? lol maybe a similar definition or a more organized one?

Answer 1

You can do a digit by digit calculation. It is slower than the 'trial and error' (Babylonian method) mentioned by others, but it has several advantages:

It can be easier for manual calculations.
Every digit found is known to be correct.
If the square root has an expansion which terminates, the algorithm will terminate after the last digit is found. It can thus be used to check whether a given integer is a square number.

The method goes as follows:

Write the the decimal expansion of the number. The numbers are laid out in a fashion similar to the long division algorithm, and the final result will appear above the original number. Divide the digits of the number into pairs, starting from the decimal point. Each pair of digits corresponds to a single digit of the result.

For each iteration:

Bring down the most significant (leftmost) pair of digits not yet used (if all the digits of the number have been used, use the digits 00) and append them to the remainder of the previous step (for the first step, the remainder is 0). This will be the "current value".

Let s denote the part of the result found so far, ignoring any decimal point (for the first step, s = 0), determine the greatest digit x such that y = (20s + x)x does not exceed the current value (20s + x is simply twice s, with the digit x appended to the right). Place the digit x as the next digit of the result.
Subtract y from the current value to form a new remainder.
If the remainder is zero and there are no more digits to bring down the algorithm has terminated. Otherwise continue with step 1.

Examples
Find the the square root of 152.2756.

...........1 2. 3 4
.........01 52.27 56

x 01 1*1 <= 1 < 2*2 x = 1
01 y = x*x = 1*1 = 1
2x 00 52 22*2 <= 52 < 23*3 x = 2
00 44 y = 2x*x = 22*2 = 44
24x 08 27 243*3 <= 827 < 244*4 x = 3
07 29 y = 24x*x = 243*3 = 729
246x 98 56 2464*4 <= 9856 < 2465*5 x = 4
98 56 y = 246x*x = 2464*4 = 9856
00 00

Algorithm terminates: Answer is 12.34

This is really hard to show here, so just go to the link below for a Wikipedia article.

Answer 2

Hey Kat, Dr. Math explains everything nicely here. I had the same curiousity and I discovered the first method of my own (guessing, dividing, and averaging) but obviously, it is a bit slow.

Dr.Math explains two of the methods which include averaging and using the binomial series expansion. The problem with averagind is that it is too slow. And the problem with the binomial series expansion is that it only works when |x| is less than one. As you can see, there is a way around it but ehhhhhhh.

The third way is to find a series which will converge at the value YOU want. And that can be done with something called the Taylor series expansion around the point x (whatever you want).

The fourth way is to write an equation and solve for the zero of that equation. One of the ways of finding roots of a function include Newton's method.

For example, if I want to find the sqrt(80).
which is actually 8.944

Then for the first method, I will
guess 2 (whatever, the better the guess, the faster it converges)
divide and I get 40
average 40 and 2 and I get 21
divide by 21 and get 3.8095
average 21 and 3.8095 and get 12.404
and keep on going until you get the accuracy you want.

For the second method, Dr.Math explains it much better.

For the third method, you need at least second semester (intermediate calculus) to actually FIND the series but if I was to find and give you the series, you could just simply add and multiply and get the answer (to any degree of accuracy).

I don't know what the actual series is (to give you an example). But if I wanted to calculate ln(1+x)

I say
ln(1+x)=x - x^2/2 + x^3/3 - x^4/4...and it goes on forever
This series only works if x is in (-1,1).

So if I want to find out ln(1.14) which actually is
0.13102826240640409278552188089983...

I use the above series AND let x=0.14 and plug x in.

Using the first four terms gives me
ln(1.14)=(0.14) - (0.14)^2/2 + (0.14)^3/3 - (0.14)^4/4
ln(1.14)=0.14 - 0.0098 + 0.000914666... - 0.00009604
ln(1.14)=0.131018626

which you can obviously see is accurate up to 4 decimal places and we ONLY used 4 terms. Now you know how your calculator can do it so fast and so accurately.

For the fourth method, we say that x=sqrt(80)
=>x^2=80
=>x^2-80=0

So square root of 80 is one of the roots of the quadratic y=x^2-80. So just find the roots, using whatever methods you know. Here you will obviously want to use a numerical method and one of the best ones (that I know of) is called Newton's method and yes, it also involves some knowledge of calculus.

Out of all of these, I say infinite series is probably the best one but you need to do some work to actually get the series before you can use it. Evaluating it is not that hard. You could easily evaluate the series once given.

Answer 3

GOOD QUESTION! I like you more than other askers already. Unless this is for an assignment, in which case, next time pay attention.

If you're not using a calulator, and you're not quite to the concept of Linear Interpolation, then your best bet is actually trial and error. Don't worry, there is a method, it just takes a little guesswork.

Suppose you want to find the square root of 55. First, pick a number to divide it by. You know that 7 is too low and 8 is too high, because 7^2 is 49 and 8^2 is 64, and 55 is between those. So, divide by 7.5. I'll let you write that out; division is hard to type. It's not too hard to see that you come up with 7.3333.

Now you know that sqrt(55) is between 7.3 and 7.5, so what will you try this time? How about 7.4?

55/7.4=7.4324. Now you know, for sure, that the first decimal place is 4, because sqrt(55) must be between 7.4 and 7.43.

You can divide again and again and get closer each time, and earn decimal places one by one. You can see that it's time consuming and gets harder and harder to gain accuracy, but for math geeks such as me, and I hope you, it can a lot of fun*!



*If you have nothing better to do. This may include gnawing your own flesh, because it gets more and more boring, too.

Answer 4

first you need to understand that aside from perfect squares (1,4,9,16,25,36,49,64,81,100,etc.), most numbers do not have square roots that can be expressed exactly in whole numbers or decimals, they can only be expressed as the multiplication of two numbers.

to find the exact square root of a number, first you need to find a perfect square factor ex. 48 is divisible by 1,2,3,4,6,8,12,16,24, and 48. Of these, 16 is the largest perfect square factor, so you must break 48 down into 16*3 and then get the square root of that which is 4*sqrt3, so the square root of 48 is 4*sqrt3

email me if you have more questions

Answer 5

Break it down into something smaller.

For example: sqrt 320

sqrt 320 = sqrt (64 x 5) since we know sqrt 64 = 8
= 8 sqrt 5

And thats about as simple as you can get without using decimals.

As far as PEMDAS is concerned: A square root is the same as a fractional exponent.

Answer 6

Depending upon how many decimal places you want, this will work:

Example: find the sqrt(650) to 2 decimal places:

First place the number in groups of 2 digits starting at the decimal point and going out in both directions...

6 50.00 00

Now if you look at the 6, the LARGEST PERFECT square that it contains is 4 (because 9 is the next perfect square and it is larger than 6).. so start off with 2 * 2 but written as in division...
....2________
2 /6 50.00 00 <---- this is written as if you were dividing 2 into 650 (but spaced out in groups of two digits.. hard to describe here...

Ok.. I put it on my web page at: http://www.tom-olsen.com/Mathematics.html

it is still hard to see, but better than what I can do in this format.

Answer 7

You use indices or what you call here in the US exponent.For example you want to find the square root of 4 which is four raised to the power half.But you know that 4 is equal to 2x2 or two squared,so the square root of four will be two squared raised to the power half.When you open your parenthesis you will have two raised to the power two times half.The two in the power will clear off with that on the half,you are left with two which is what your calculator will give you.
Always break down the number into bases of twos or threes or fives.

Answer 8

if you square a number,lets say 8, thats 8 x 8= 64..the square root of 64 is 8, sounds easy, lol..the square root of 16 is 4, got it?

Answer 9

Continuously divide the number by two until you get to 0. The number of times you can do this is the square root of the number.

Answer 10

Square root first, than exponets.

Wait, Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction. Use the square root before you do anything else.