How do you find the square root of 166 without a calculator?

Answer 1

Bear with me. This is kind of complicated, but it works.

The first step is to break the number up into pairs of number from both sides of the decimal.

Start with √1'66.00'00'00'00 and work from there.

This is now a modified long division problem. You need to find the new divisor each time, but it is only twice the current answer, plus a new digit.

First, find the lowest square less than or equal to the first block

In this case, it is 1, so the current answer is 1. Subtract 1 from the one in the first block, and 1-1=0 so it goes away

Now, bring down the 66, and double the current answer (2*1=2) and bring that down as the partial divisor. This is where it gets a little tricky.

You have 66 divided by 2- You need to put a number in there to make twenty-something, but the number you add is the new digit in the answer, so will be multiplied.

1 would make 21*1=21 and be too small since 66 - 21 = 45 > 21

2 could work, since 22 * 2 = 44 and 66 - 44 = 22

3 is too big since 23 * 3 = 69

Now you have the next digit, and your answer so far is 12

Take 66-44=22 and bring down 2 more digits, the two zeroes next.

Your new dividend is 2200, and bringing down double the answer gives 24-

What number can do there to make sense? Since the 22 remainder last time was high, try 9

249 * 9 = 2241. too high

248 * 8 = 1984. 2200 - 1984 = 216. good.

Current answer is 12.8. bring down 2 more digits.

New dividend is 21600, and double the current answer is 128*2=256

What number can you multiply by 256- to get close to 21600? 7?

2567 * 7 = 17969. Too low.

2568 * 8 = 20544. Nice. 21600 - 20544 = 1056. bring down 2 more digits.

dividend: 105600 Double current answer is 2576

Looks like 4

25764 * 4 103056. very nice! 105600 - 103056 = 2544

2 more digits makes the divisor 254400, but the new divisor is 25768- and even 257681 is too high, so the next digit in the answer is 0. Bring down 2 more digits and go on.

dividend: 25440000, and divisor is 257680-

257689 * 9 = 2319201

I think that this is enough working out to show you how it works, and give you enough digits in √166 ≈ 1.88409

Any number can be done this way. Just remember the doubling of the current answer and tacking the next digit in the answer to the end of the divisor.

The doubling is somehow related to the fact that (x+y)² = x² + 2xy + y²

Enjoy, but practice it until you feel comfortable with it. ■