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
(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.
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.