Squaring numbers close to multiples of base 10, 100, 1000 and so on

To square numbers close to the base of multiples of 10, 100, 1000 and so on follow the steps below:

Step-1:    Find the surplus or deficit from the working base of multiples of 10, 100, 1000 and so on.

Step-2:    Add the surplus (if it is more than base) or subtract the deficit (if it is less than base) from the whole number given.

Step-3:      Multiply the result by the multiples of 10, 100, 1000 and so on the number is.

Step-4:     Find the square of surplus or deficit and write the result in the last places. Be sure that, for numbers multiples of 10; 1 more digit to go, since 10 has 1 zero, for numbers multiples of 100; 2 more digits to go, since 100 has 2 zeros, accordingly since 1000 has 3 zeros, 3 more digits to go for numbers multiples of 1000 and so on. So, carry forward or put extra zero(s) if necessary to place the digits accurate.

408² =?

Notice that 408 is close to 400 which is 4 multiples of 100. So, our working base is 400 and actual base is 100.

1)       Find the surplus or deficit from base 400. It is (408–400=) 8 surpluses.

2)       Since 408 is 8 more than base 400 so, add 8 to the given number, which is 408+8 = 416.

3)       Now, since our working base 400 is 4 multiples of 100; multiply 416 by 4 to get 1664. Write 1664 as the first part of the answer.

4)       Now, as the actual base 100 has 2 zeros, so 2 more digits to go. Find the square of surplus 8 which is 64. Write 64 as the last 2 digits of the answer.

So, our final answer is 166,464.

588² =?

Notice that 588 is close to 600 which is 6 multiples of 100. So, our working base is 600 and actual base is 100.

1)       Find the surplus or deficit from base 600. It is (600–588=) 12 deficit.

2)       Subtract 12 from the given number, which is (588-12=) 576.

3)       Since working base 600 is 6 multiples of 100; multiply 576 by 6 to get the first part of the answer which is (576×6=) 3456.

4)       Square of 12 is 144. Now, as the actual base 100 has 2 zeros, so 2 more digits to go. So, write 44 as the last 2 digits of the answer and carry 1 forwarded to 3456 to get 345744 as the final answer.

So, 588² = 345,744

Find 67²

67 is close to 70 which is 7 multiples of 10. So, our working base is 70 and actual base is 10.

67 is (70 – 67=) 3 deficit from the base.

Since it is less than base, subtract the deficit from the number i.e. 67 - 03 = 64.

Multiply this result by 7 since the base is 7 X 10 = 70.

64 x 7 = 448

Square of deficit = 3² = 9.

So the answer is 4,489.

5015² =?

Nearest base is 5000. Surplus = 15 and 5000 = 5 x 1000.

Since it is more than base, add the surplus to the number to get (5015 + 15 =) 5030.

Multiply this result by 5 since base is 5 X 1000 = 5000.

5030 x 5 = 25150

Square of surplus = 15² = 225

So the final answer is 25,150,225 [since we have taken multiples of 1000, write down 225 as it is].

789²=?

Actual Base is 100 and working Base is 800.

Deficit from working base is (800-789=) 11.

Subtracting the deficit from the number gives (789-11=) 778.

Since 800 is 8 times 100, multiplying 778 by 8 gives (778X 8=) 6224.

Squaring the deficit gives (11² =) 121.

Since the base has 2 zeros, so place 21 and carry forward 1 to 6224 to get 6225.

So the final answer is 622,521.