| FINDING THE SQUARE ROOT OF LARGER NUMBERS ... |
That's a good question and I'm glad you asked it! In our previous examples, we were only finding square roots of number less than 100 because they are easily found on the A or B scales. I think it's time we had a closer look at the A and B scales - Care to join me? If you look very carefully at the A scale and D scale, you will see something quite amazing! The A scale appears to be two D scales (or two C scales, remember the C and D scales are the same too) joined together. If you thought that, then you're right. Let me shrink the D scale and see how it compares with the first half of the A scale ... 
Wow! Check it out! They're the same! Well that's neat, but why do two D scales make up the A scale? The reason for this is due to the mathematical relationship between the logarithm of the square of a number and the number. To get a bit technical, the logarithm of the square of a number is twice the logarithm of the number itself, so we have two D scales making up the A scale and believe it or not, with just these two smaller D scales, we can find the square root of any number! Don't worry too much about what a logarithm is yet, just remember that the slide rule is based on many mathematical rules and relationships. How do we find the square root of larger numbers? When we are finding the square root of numbers we need to be careful which half of the A scale we are going to use. As you might recall from the previous examples, if we wanted to find the square root of a number between 1 and 10, we used the first half and if the number was bigger than 10 and up to 100, we used the second half. So this tells us that the two halves are used for different ranges of numbers. Now I told you that we could find the square root of any number, so how do we do that? Suppose I want to find √400 (you might already know what it is). When we look at our A scale, it only goes from 1 to 100, so we have to do something first. We need to write the number 400 in a different way ... 4 x 100 to be exact. The reason we do this is because it is very easy to find the square root of 100 - we know that it's 10. So we could write √400 like this 
So I know that to find √400 is the same as finding the √4 and multiplying by 10. Since the number 4 is in the first half of the A scale, I will move the cursor to 4 to find its square root. 
We can now calculate the √400 ... √400 = √4 x 10 = 2 x 10 = 20
That wasn't too bad, but what about something like √400000? Once again, we need to write 400000 in a different way ... 
But hold on ... I could go a bit further ... 
See how we can take out all factors of 100, I can now see that √400000 is equal to √40 x 100. On our slide rule 40 is in the second half of the A scale, so I move my cursor to the 40 in the second half and read what √40 is. 
Now I can calculate √400000 to be approximately ... √400000 = √40 x 100 ≈ 6.32 x 100 = 632 You may think that there is an easier way to do this and ... you're right! What we do is count how many pairs of digits there are from the right until we arrive at a number between 1 and 100. Let me show you how to do this. Let's say we want to find √63000. I start at the right and mark of pairs of digits ... 
I can go one step more ... 
I don't want to mark off any more pairs because there is only one digit in front. For each pair that I mark off, I will be multiplying my final answer by 10. So in this case, I will multiply my final answer by 10 x 10 = 100. To find √63000, I would find √6.3 from my slide rule using the first half of the A scale and then multiply that answer by 100. 
As my final step I can now approximate √63000 as ... √63000 = √6.3 x 1000 ≈ 2.51 x 100 = 251
Got it yet? Let's try one more and then you can do some practice. How about √78000000? Now this seems harder!
√78000000 = √78 x 1000 ≈ 8.83 x 1000 = 8830
Okay, it's your turn now ... off you go and get some of that valuable practice by clicking here!
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"Finding square roots was very easy, but what do I do if the number is bigger then what's on the scales?"
