Finding Perfect Squares

Now if we find a number such as √225 we know that it is 15 if we memorized our tables.

What if we run into a very large number like 21316?  If we were told find the root without a calculator, it would be very hard you would think.  We need to come up with a quick formula for solving this.

Lets break this down into a formula.  We will create an equation for the root by placing A in the 10's place and B in the units place.  We will square this number to come up with an equation for solving the root.

10A+B= root
(10A +B)²=21316

100A² + 20AB + B²=21316

Finding A:

• Before finding A, I will subtract 16 from 21316: 21316-16=21300
• Divide 21300 by 100: 21300÷100=213
• We will find the nearest perfect square of a number less than 213 and let this number =A.  That would be 14 because 14²=196.  We cant use 15 because 15²=225 which is larger than 213.  So A=14.

Finding B:

• We will take 100A²=100X14²= 19600 and subtract it from or original number:  21316-19600=1716
• Our equation becomes 20AB+B²=1716 or if we substitute A back into our equation we have 20X14XB+B²=280B+B².  Because the number ends in a 6 we know that the units digit or B is either 4 or 6.  4 does not work because 280(4)+4²=1136≠1716.  If we substitute 6 into our equation, we have 280(6)+36=1716.

Voila our number is 156.  15 is in the tens place and 6 is in the units place.

It is not really that hard.  We will do an easier number 841.

If we subtract 41 and divide by 100, we have 8 and we have 2²=4 which is the largest perfect square less than 8. A=2 so 100A²=400.  20AB+B²=841-400=441.  So 20AB=20(2)B+B² =40B+B²=441.
B can be either 1 or 9.  B is 9.  The square root of 841 is 39.