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Using the Ones Digit to Estimate Square Roots

Date: 02/11/2005 at 11:56:25
From: Tina
Subject: How to find square roots

Dear Dr. Math,

I am having trouble figuring out square roots quickly.  For bigger
numbers like 289, I have to try a lot of numbers before I get the 
answer.  Usually, I square a number near 20, and see if the result is
less or greater than the squared number.  Then I try another number.

Could you please help me?
         
Sincerely,

Tina



Date: 02/11/2005 at 13:38:35
From: Doctor Ian
Subject: Re: How to find square roots

Hi Tina,

Just looking at the units place can tell you a lot.  For example, with
289, we have to multiply something so that the units digit squared
ends in a 9.  The only way for that to happen is for the units digit
to be a 3 or a 7.  So you just have to check 13 and 17.

Similarly, suppose you want to find the square root of 256?  The units
digit is 6, which means the square root has to have a units digit of 4
or 6, so you only have to check 14 and 16.  

There aren't all that many possibilities for the units digit:

  _0^2 = _0
  _1^2 = _1
  _2^2 = _4
  _3^2 = _9
  _4^2 = _6
  _5^2 = _5
  _6^2 = _6
  _7^2 = _9
  _8^2 = _4
  _9^2 = _1

So looking at it the other way, 
    ___
  \/ _0 = _0^2

    ___
  \/ _1 = _1^2 or _9^2

    ___
  \/ _4 = _2^2 or _8^2

    ___
  \/ _5 = _5^2

    ___
  \/ _6 = _4^2 or _6^2

    ___
  \/ _9 = _3^2 or _7^2 


Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Square Roots

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