Question

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How to find the square of a 3 digit number easily??





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Answer 1

Step-by-step explanation:

Ignore the hundreds digit and square the tens and unit digits using the method for squaring 2 digit numbers.

Do an "open cross-product", where you multiply the first and last digits then double the result.

On the Hundreds and tens digits do another squaring 2 digit number but this time omit the first step of squaring the units digit.

Example 1 : 6782

To illustrate what we are doing we will use squaring 678 as our example.

Squaring 3 digit numbers-678-01

Step 1

To begin our solution we will use the method for squaring 2 digit numbers on the tens and units digits, the 7 and 8. In this step we treat the 78 as a two digit number and we ignore the 6.

Squaring 3 digit numbers-678-02

a) square the units digit.

The 8 is our units digit so we square that giving us a result of 64.

Squaring 3 digit numbers-678-03

We write down the 4 and carry the 6.

Squaring 3 digit numbers-678-04

b) multiply the two digits and double.

We multiply the 7 and 8 together then double the result. We get 112 from this to which we add the carry from a for a total of 118.

Squaring 3 digit numbers-678-05

We write 8 and carry the 11.

Squaring 3 digit numbers-678-06

c) square the tens digit.

The 7 is our tens digit and 7 squared is 49. We add 11, the carry from b, to the 49 giving us 60.

Squaring 3 digit numbers-678-07

We write the 60 in front of the 84 we already have giving us our initial four digit number.

Squaring 3 digit numbers-678-08

Remember, if the result is less than four digits then put leading a zero to make it four digits.

The tens and units of this answer, the 84 are, in fact, the final result for these digits. The remaining two steps will only affect the hundreds and up.

Step 2

We now do our "open cross-product" by multiplying the first and last digits in our three digit number, the 6 and the 8. We then double the result.

Squaring 3 digit numbers-678-09

Squaring 3 digit numbers-678-10

The trick here is you add the result of this step to the first two digits of the result of step 1, which is why step 1 must have a four digit result. Or, to put it another way the units of the result of this step is put in the hundreds column.

Squaring 3 digit numbers-678-11

There are two ways you can proceed here, add the numbers up in each step and get a progressive total or simply write the result of each step in its correct place then column-wise add up everything at the end. For clarity I will take the second option and write everything in its place. If you were mentally doing this then the progressive total approach would be better.

Step 3

We now use the method for squaring 2 digit numbers on the hundreds and tens units, the 6 and the 7, but this time we will omit the squaring of the units. Again we treat the 67 here like a two digit number and ignore the 8.

Squaring 3 digit numbers-678-12

a) multiply the two digits and double.

We multiply the 6 and the 7 together then double the result.

Squaring 3 digit numbers-678-13We write the 84 so that the units digit, the 4, is under the tens digit, the 9, from step 2.

Squaring 3 digit numbers-678-14

b) square the tens digit of our two digit number.

The 6 is the tens digit of our two digit number so we square that giving us 36.

Squaring 3 digit numbers-678-15

We put the 36 so that the units digit, the 6, is under the tens digit, the 8 from part a.

Squaring 3 digit numbers-678-16

We now have all the numbers we need and they are all lined up correctly so all we need to do is to add them up column-wise. If we start on the right and work left we only have a 4 in the first column.

Squaring 3 digit numbers-678-17

In the next column we only have an 8.

Squaring 3 digit numbers-678-18

In the third column we only have a 6 and a zero so the sum is 6.

Squaring 3 digit numbers-678-19In the fourth column we have 6, 9 and 4, adding these together we get 19 so we write 9 and carry 1.

Squaring 3 digit numbers-678-20

In column 5 we have 8 and 6 to which we add the 1 from the carry for a sum of 15. We write 5 and carry the 1.

Squaring 3 digit numbers-678-21

In the last column we just have 3 to which we add the 1 we carried, for a sum of 4.

Squaring 3 digit numbers-678-22

We now have our result of 459,684 as the square of 678.

As I said before you don't have to write everything out as I have done, but you could if you feel more comfortable doing so until you get more familiar with the method. See the fourth example as to how you can just write down the answer

Answer 2

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Here we can see that 999 is not a perfect square as it leaves 38 as remainder. So if 38 will be addede to 999 then it will become a 4 digit number. Therefore to find the largest 3 digit perfect square we will subtract 38 from 999. Hence 961 isthe largest 3 digit perfect square whose square root is 31.