How do you calculate manual roots (square, cubic, etc.)?

Question

For calculations on the Physics GRE

Answer 1

A good method to remember is newton's method: Suppose you have a differentiable function f(x) that you want to find the root of. Then let x be a guess as to the root. You can get a more accurate guess X using the forumula: X=x-f(x)/f'(x). This method can be iterated, and converges quadratically to the correct root. So to find the square root of n, you let x²-n=0 and find x using X=x-(x²-n)/2x=x-x/2+n/2x=(x+n/x)/2. For the cube root, you let x³-n=0, and so your formula is X=x-(x³-n)/(3x²)=x-x/3+n/3x²=(2x+n/x²)/3. This can be generalized to any n.

Example: √2

x0=1
x1 = (1 + 2/1)/2 = 1.5
x2 = (1.5 + 2/1.5)/2 = 1.41666...
x3 = (1.41666... + 2/1.41666...) = 1.414215...

Answer 2

If you have a calculator, use Newton's method. Suppose you want to take the nth root of A. Guess the answer, say x. Compute a better guess y by setting y = ((n-1)x-A/x^(n-1))/n. This is a weighted average of x and A/x^(n-1) using weights of n-1 and 1. Now let x = y and compute a new y. Keep doing this, probably no more than 3 times and usually no more than 2, until you get enough digits of precision. Convergence is fast.

If you don't have a calculator, then you are limited to square roots. To do those, try square root of 47. Set up the problem and take the integer nearest the square root of 47.

47.00000000 |6
-36
1100

Subtract the square of your guess and drop the next two digits.

Multiply the result by 2 and get 12. Divide 12 into 1100 and get 8. (there is a digit after the 12 that is not yet written in). Add the 8 to the 12 to get 128. Then multiply the 128 by 8 and subtract:

47.00000000 |6 (6)
-36
_1100 |128 (68)
_-1024
____7600 |136

Now divide the 7600 by 136*, and get 5. Keep on going like this as far as you want. If you had used 9 above, the subtrahend would have been 1161, which is too large. The result is ~6.85.

There is a similar, much more complicated procedure for cube root. Both are based on the binomial formula. Do a quartic root by doing two successive square roots. Beyond that, forget this method.

Hope these can be of help to you. The second method may be hard to find since it is so outmoded. I learned it by myself when I was a child by reading my father's correspondence school texts. It never appeared in my grade or high schooling nor did I encounter it in college.

Answer 3

Method 1:

Start with the number you want to find the square root of. Let's use 12. There are three steps:

Guess
Divide
Average.
... and then just keep repeating steps 2 and 3.

First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12.

In step two, we divide 12 by our guess of 2 and we get 6.

In step three, we average 6 and 2: (6+2)/2 = 4

Now we repeat step two with the new guess of 4. So 12/4 = 3

Now average 4 and 3: (4+3)/2 = 3.5

Repeat step two: 12/3.5 = 3.43

Average: (3.5 + 3.43)/2 = 3.465

We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing.

3.465 * 3.465 = 12.006225

That is quite close to 12.

Method 2:

Find the square root of 676.

Step 1 - Starting at the decimal point, of the number under the square root symbol, separate the digits into groups of two.
Step 2 - Locate the decimal point of the answer.
Step 3 - Find the largest square (4) that subtracts from the left-most pair (6) and still yields a positive result. Write it under the first group.
Step 4 - Write the square root (2) of this largest square (4) above the first group, as the first figure of the square root.
Step 5 - Subtract the square number from the first group. This is the remainder (2) that will be used in the next step.
Step 6 - Annex the next pair of digits with the remainder to form a dividend (276).
Step 7 - Multiply the square root developed so far (2) by 20 to form a trial divisor (40).
Step 8 - Divide the dividend (276) by the trial divisor (40). Add the result to the root already found. Also, add the result to the trial divisor to form the complete divisor (46).
Step 9 - Multiply the complete divisor (46) by the new figure of the root (6).
Step 10 - Subtract this product (276) from the dividend (276).

Answer 4

Drop D is almost right, but you can apply a little more systematic method than random guesses. The folks that mentioned Newton's method are exactly right, but overlook the fact that if you had a calculator, you wouldn't need to guess.

Use the Newton-Raphsonn Method.

You guess a number. However far off it turns out to be, divide by the derivative of your function. Subtract that from your first guess and use the result as your second guess. You repeat until you get to the necessary accuracy.

For example, if you want the cube root of 1729.03, you first set up your equation:

x^3 - 1729.03 = e (e will tell you how far you missed by)

Using the power law, the derivative of x^3 - 1729.03 is 3x^2.

I know the answer is greater than 10, since to cube 10, you just multiply the number of zeroes by 3 to get 1000. I know it's higher, but I don't know how much higher. 11 or 12 will make a pretty good first guess (it doesn't matter if you guess too high or too low, but getting pretty close cuts down the number of guesses you have to make). I'll pick 11.

11^3 - 1729.03 = 121*11 - 1729.03 = 1331 - 1729.03 = -398
Two things to note. If I'm doing this in my head, I'm going to round that 398 off to 400 to make things easier. If you notice what happened to 11 as we progressed up the powers, you'll see that 11 was chosen as much because it's easy to multiply as because I thought it might be closer than 12.

The derivative is 3 * 11^2 = 3 * 121 = 363

My second guess should be 11 - (-400/363). That's approximately 11 + 10/9 = 12.1

12.1^3 is pretty hard. Hopefully 12.1 is close enough. Since I fudged my denominator more than the numerator, the answer is probably between 12.0 and 12.1.

However, 12^3 = 144 * 12 = 1440 + 288 = 1728. Wow, that's close. I'm off by only 1.03. The right way to adjust that guess would be:

-1.03/432 = -1/400 = -.25 x 10^-2 = -.0025. (432 = 3 * 12^2 which is the derivative of my function) That means 12.0025 would be just very slightly high (once again, I fudged the denominator more than the numerator). I think that's about as accurate as I could get in my head, since trying to guess just how much higher than the actual answer my guess was would be pretty hard and I don't want to be doing 12.002 in my head. Checking my answer on a calculator, I find the actual answer is 12.00238+, so my guess was off by less than .00012.

Square roots are easier. The derivative is 2x.

The main point is that you have a general rule to follow in how to narrow your guesses down to the right answer very quickly. It's only a general rule. A good feel for numbers allows you to fudge your revisions just a little without slowing your progress down. You'll still wind up off just a little, but the extra accuracy just isn't usually worth sitting down and figuring these out with a pencil and paper.

Edit: And, yes, for all of you Feynman fans, I did know that 12^3 was 1728, but the point is that you'll still get there even with a 'bad' first guess.

Answer 5

I know with square roots you do this:
The square root of 9 would be 3 because 3 x 3 = 9. So, doubles come out. Like the square root of 4 is two.. Cause 2 x2 =4. and with for instance, the square root of 3136 is 56 because 7 x 7 x 8 x 8 = 3136, therefore the two 7's come out as one 7. and the two 8's come out as one 8. Then you multiply 7 x 8 = 56. Voila!

Answer 6

Good old trial and error. For example, what is the square root of 14? Let's see...3 squared is 9, and 4 squared is 16. The answer must be between 3 and 4. Try 3.5 squared. If too low, try 3.6. And so forth. It is tedious, but GRE is multiple choice, so you can narrow down the answer.

Answer 7

Hi:

Try the following websites for this:

http://en.wikipedia.org/wiki/Cube_root
http://en.wikipedia.org/wiki/Square_root
http://mathworld.wolfram.com/CubeRoot.html
http://mathworld.wolfram.com/SquareRoot.html
http://dic.blogopt.com/Shifting_nth-root_algorithm
http://dic.blogopt.com/Radical_%28mathematics%29

This one is for pencil and paper or schoolboy method:

http://www.nist.gov/dads/HTML/cubeRoot.html
http://www.nist.gov/dads/HTML/squareRoot.html
http://www.qnet.fi/abehr/Achim/Calculators_SquareRoots.html


And for roots that multiple of 2 just do the square root that multiple

For example to get the 8th root of 2 just do the square root three times


Other way is is take the logarithm of the number being rooted and dividing it by the root number and take the result and antilog it

for example 5th root of 10

first find the logarithm of 10 is 1 {base 10 } and divided by five with is .2

Antilog ( .2)= 1.584893192...

So the 5th root of 10 is 1.584893192..

You can use a Mathematical Formula and table Book to look up the logarithm for that

The only other way is to use the Newton method :


gn= ( ( N / gn ^ (r-1 ) ) + ( r - 1 ) *2 ) ) / r

gn = guess number or { number rooted if at the end of program run

N= Number to be rooted

r= root number

repeat using the new calulated guess number each time

For example


cube root of 6

run # 1

gn = 1

gn= (6 / 1^ (3-1))+ (3-1)*1)/3 [ to save time (3-1) is replaced by 2 okay]

gn= (6/ (1^2) + (2*1))/3

gn = (6+2)/3

gn=7/3

gn = 2.33333...

end of run # 1

start of run # 2:


gn= 2.3333

gn = ((6/(2.333333...^2))+(2*2.333))/3

gn = (6/5.444..) + 4.66666) /3

gn = (1.10204040906 + 4.6666666666)/3

gn = 1.922902494

end of run # 2

Strarting Run # 3:

gn = 1.922902494

gn = (6/(1.922902494^2))+(2*1.922902494))/3

gn = ((6/ 3.697554002) + 3.845804988)/3

gn = (1.622694353+ 3.845804988)/3

gn = 5.468499342 / 3

gn= 1.822833114

end of run # 3

Start of run # 4:

gn = 1.822833114

gn = (((6/1.822833114^2)+(2*1822833114))/3)

gn = (( 6/3.322720561)+( 3.645666228))/3

gn = ( (1.805749202)+ 3.645666228)/3)

gn = 5.45141543 / 3

gn= 1.8171384766

end of run # 4

Start of run # 5

gn =1.8171384766

gn = ((6/(1.8171384766^2)+( 2*1.81713847666))/3

gn = ((6/ 3.301992243)+ 6.603789504))/3

gn = (1.817084825 + 5.4513617783)/3

gn= 5.451361779 /3

gn = 1.81712059

end of run 5

the cube root of 6 is 1.8172059 { accurate to four decimal places}

Continue running it some more to increase the number of decimal place accuracy. I usually run this formula about six to ten times for really good results ( I get ten to twelve digit decimal accuracy by doing this way. )

Answer 8

I'll tell you what I remember for square roots.

It is a lot like a division but:

1. You take two digits at the same time
2. You start from the end of the number.
Hope this helps(at least a little bit)