How do you get square roots in long way?

Answer 1

Below are two methods of working out a square root. The first is simpler, but takes longer to give you an answer and to increase the accuracy of it. The second, although slightly more involved, wil do the job quicker on average.

Start with a number you know the square root of, as close to the number you are trying to do. Then gradually move upwards (or downwards, if you started above) until you get to a number that is too high. Then go back to the last number that worked, and move to the next decimal point, and start again.

For example:

We want to calculate the square root of 421. We know that 202 is 400, but that 212 is 441. So we will start at 20, and start with increments - gaps we move in - of 0.1.

Beginning with 20.1, we get 404.01; 20.2 gives us 408.04.

Up to 20.5, to make the example quick, we see that it squared gives 420.25; with a bit of experience and comparing the rate it would have gone so far, we would normally skip to just doing the next decimal point. But just to check, 20.6 would have given us 424.36, which is of course above our number of 421.

So onwards. Next, we start with 20.5, and go in increments of 0.01. 20.51 gives us 420.6601; again, this is pretty close to what we need. 20.52 gives 421.0704, so we have reached the target quicker this time.

Using increments of 0.001, we would do a lot of work, but soon reach the figure of 20.518.

If we wanted to increase the accuracy of our square root, we could now use increments of 0.0001, but for most purposes what we have already done is enough. You could try to finish the example if you like, and use a calculator to check it.

To 2 decimal points and 4 significant figures, the square root of 421 is 20.52.

This is a little confusing at first, so please get paper and work it out.

First write the number you are taking the square root of, including the decimal point, and put the same kind of long division notation over the number so you can put the answer above it. Next, mark off two digits at a time, going left and right of the decimal., and write the decimal point in the same place. To use the example above, you would write it as 4'21.00'

Now, take the highest perfect square less than or equal to the first section. In this case, the 4. Four is the nearest perfect square. 2 is the root of that, and that begins the answer. Put the 2 above the 4 in the answer.

Subtract just like long division, 4-4=0, and bring down the next two digits. This is why you need to mark them off at the beginning.

In this case, you are bringing down the 21.

This is where it gets a little hard.

What is the answer SO FAR? The answer so far is 2. Double that (4) and bring it down to the left of the 21 as a "trial divisor."

You are going to be dividing the 21 by "4" and another digit, and the final digit of the divisor is going to be the next digit in the answer, so ask yourself how many times 40-something goes into 21. The answer is 0.

Write that zero as the next digit in your answer. Your answer so far is 20.

Just like long division, you multiply the divisor "40" by this new digit in the answer "0" and subtract 21-0=21

Bring down the next two digits, in this case "00"

The current answer is "20" so double it "40" and write that to the left to the 2100

How many times will 400+? go into 2100...

It could be 401*1, 402*2, 403*3, 404*4. See how the last digit is what you multiply by.

405*5=2025, and that is the closest.

Write the 5 at the end of the divisor, and as the next digit in the answer.

The answer so far is 20.5

write the 2025 under the 2100 and subtract to get 75. Bring down the next two digits

Now you have 7500 at the bottom, 20.5 as the current answer. Double the 205 to get 410 and write that as your new "trial divisor"

What digit multiplied by 410? will get close to 7500? 1, because 4102*2=8204 and that is too high. Your answer so far is now 20.51

Write the 1 in the answer, and 4101*1=4101 goes under the 7500.

Subtract 7500-4101 to get 3399, bring down two more digits to make the number at the bottom 339900.

Double the answer so far and write 4102 as the trial divisor.

How many times will 4102+? go into 339900, well 4 goes into 33 eight times, so check that. 41028 * 8 = 328224 and that is close without going over.

Write the 8 as the new digit in the answer, subtract 339900-328224, bring down two more digits, double the current answer, find the new digit, and go on as long as you need for all the digits you want. So far the answer is 20.518

Yes, it is time consuming, paper consuming, frustration consuming, but it will definitely give you the exact answer without any guesswork (other than that you have to use in long division, like is the next digit 8 or 9? Try 9, and it's too high, so go with 8)