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Algebra/Square Roots


How do I find the long-hand way to figure out square and cubed roots? I mean I know cube root of 27 is 3 off the top of my head, and square root of 81 is 9, but I have no idea how that is figured out other than memorizing it from somewhere.

If we want the square root of, say, 2314, we divide it into 2-digit sections.
This gives 23 14.  The square that is less than 23 is 4, for 4^2 = 16.
Subtracting that off gives 23 - 16 = 7, so next look at 714.

Take 2*(what we have so far), which is 4, and get 8.
Find x where eighty-x times x is less than 714.
It can be seen that 88*8 = 704, so the next digit is 8.
This makes 48 the number right now, and 48^2 = 2,304.

To find the next digit, take the difference, which is 714 - 704 = 10.
Add two more zeros to that and get 1000.
The number we have so far is 48, and 2*48 = 96.
Find nine hundred and sixty x times x, and we can see that x = 1.
Subtract of 961x1 = 961 and get 39.  Add two more zeros to get 3900 and repeat the process.

Thus, so far, we have 48.1.
Looking at 48.1^2, it is 2314.61, which is still less than 2314.

The next step would be taking 962x times x where the result is less than 3900.
It looks like this one is a 0, for nine thousand six hundred and twenty one times one
is too large.  So far, we now have 48.10.

As far as cube roots, I never did remember how to do them, but did read it can be done.  It took a several page explanation in how to do them.


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Scott A Wilson


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