1. Prime Factorization Trick: Find the prime factors of the given number and group them in pairs. Take one number from each pair and multiply them together. This product will be the square root.
2. Estimation Trick: Approximate the square root by finding the nearest perfect square less than the given number. Then, take the square root of this perfect square and adjust it based on the difference between the perfect square and the given number.
3. Digit-by-Digit Trick: Separate the given number into groups of two digits from right to left. Find the largest digit whose square is less than or equal to the leftmost group. This digit becomes the leftmost digit of the square root. Repeat this process for the remaining groups.
4. Number Line Trick: Draw a number line and place the given number on it. Start with an initial guess for the square root and move towards the given number, adjusting the guess iteratively until you reach a close approximation.
5. Babylonian Method: Start with an initial guess for the square root and repeatedly improve it by averaging the guess with the given number divided by the guess. Repeat this process until the guess converges to a close approximation of the square root.
These tricks can help you find square roots using different approaches. Experiment with them to see which one works best for your needs!
ab2 = left most part| middle part | right most part
During calculations, we shall pass from the leftmost duplex to the leftmost duplex.
The left most part will be duplex of ‘a’, the middle part will be duplex of ‘ab’, finally the right most part will be duplex of ‘b’.
i.e ab2 = D(a)| D(ab) | D(b)
= a2 | 2ab | b2
[toggles type=”accordion”]
[toggle title=”Example 1: Find the square of 12″]
1$^2$ = D(1)| D(12) | D(2)
= 1$^2$ | 2x1x2 | 2$^2$
= 1 | 4 | 4
Hence, 12 = 144 using the duplex methodology
[/toggle]
[toggle title=”Example 2: Find the square of 23″]
23$^2$ = D(2)| D(23) | D(3)
= 2$^2$ | 2x2x3 | 3$^2$
= 4 | 12 | 9
Here, the middle portion has more than two digits, please note that only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number ctively.
Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 12 to 13 and for 13 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 9 to 10.
Hence,
321$^2$ = 10 | 3 | 0 | 4 | 1
= 103041
Example 5: Find the square of 791
7912 =D(7) | D(79) | D(791)| D(91) | D(1)
= 7$^2$ | 2x7x9 | 2x7x1 + 9$^2$ | 2x9x1 | 1$^2$
= 49 | 126 | 95 | 18 | 1
= 625681
[/toggle]
[/toggles]
To practice two digit squares, check out our exercises in Three Digit Squares
Square of Four Digit Numbers
Proceeding on the same methodology as for 2 and 3 digit squares as mentioned above, to quickly square a 4 digit number, we must know duplex of 4 digit number as well.
For a number with four digits, say ‘abcd’, Duplex of ‘abcd’ => D(abcd) = 2ad + 2bc
The square of ‘abcd’ will have seven parts as shown below
As mentioned above only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number preceding the units digit, to the immediate left part , and add it there respectively.
Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 8 to 9
Answers & Comments
Verified answer
Answer:
1. Prime Factorization Trick: Find the prime factors of the given number and group them in pairs. Take one number from each pair and multiply them together. This product will be the square root.
2. Estimation Trick: Approximate the square root by finding the nearest perfect square less than the given number. Then, take the square root of this perfect square and adjust it based on the difference between the perfect square and the given number.
3. Digit-by-Digit Trick: Separate the given number into groups of two digits from right to left. Find the largest digit whose square is less than or equal to the leftmost group. This digit becomes the leftmost digit of the square root. Repeat this process for the remaining groups.
4. Number Line Trick: Draw a number line and place the given number on it. Start with an initial guess for the square root and move towards the given number, adjusting the guess iteratively until you reach a close approximation.
5. Babylonian Method: Start with an initial guess for the square root and repeatedly improve it by averaging the guess with the given number divided by the guess. Repeat this process until the guess converges to a close approximation of the square root.
These tricks can help you find square roots using different approaches. Experiment with them to see which one works best for your needs!
Math Trick to find Square of Two Digit Numbers:
Consider a general two digit number, say, ‘ab’.
The square of ‘ab’ will have three parts.
ab2 = left most part| middle part | right most part
During calculations, we shall pass from the leftmost duplex to the leftmost duplex.
The left most part will be duplex of ‘a’, the middle part will be duplex of ‘ab’, finally the right most part will be duplex of ‘b’.
i.e ab2 = D(a)| D(ab) | D(b)
= a2 | 2ab | b2
[toggles type=”accordion”]
[toggle title=”Example 1: Find the square of 12″]
1$^2$ = D(1)| D(12) | D(2)
= 1$^2$ | 2x1x2 | 2$^2$
= 1 | 4 | 4
Hence, 12 = 144 using the duplex methodology
[/toggle]
[toggle title=”Example 2: Find the square of 23″]
23$^2$ = D(2)| D(23) | D(3)
= 2$^2$ | 2x2x3 | 3$^2$
= 4 | 12 | 9
Here, the middle portion has more than two digits, please note that only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number ctively.
Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 12 to 13 and for 13 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 9 to 10.
Hence,
321$^2$ = 10 | 3 | 0 | 4 | 1
= 103041
Example 5: Find the square of 791
7912 =D(7) | D(79) | D(791)| D(91) | D(1)
= 7$^2$ | 2x7x9 | 2x7x1 + 9$^2$ | 2x9x1 | 1$^2$
= 49 | 126 | 95 | 18 | 1
= 625681
[/toggle]
[/toggles]
To practice two digit squares, check out our exercises in Three Digit Squares
Square of Four Digit Numbers
Proceeding on the same methodology as for 2 and 3 digit squares as mentioned above, to quickly square a 4 digit number, we must know duplex of 4 digit number as well.
For a number with four digits, say ‘abcd’, Duplex of ‘abcd’ => D(abcd) = 2ad + 2bc
The square of ‘abcd’ will have seven parts as shown below
abcd2 = D(a) | D(ab) | D(abc) | D(abcd)| D(bcd) | D(cd) | D(d)
= a2 | 2ab | 2ac + b2 | 2ad+2bc | 2bd + c2 | 2cd | d2
Example 6: Find the square of 1221
1221$^2$= D(1) | D(12) | D(122) | D(1221)| D(221) | D(21) | D(1)
= 1$^2$ | 2x1x2 | 2x1x2 + 2$^2$ | 2x1x1+2x2x2 | 2x2x1 + 2$^2$ | 2x2x1 | 1$^2$
= 1 | 4 | 8 | 10 | 8 | 4 | 1
As mentioned above only the leftmost part can have more than one digit. For the rest of the parts, we need to carry over the number preceding the units digit, to the immediate left part , and add it there respectively.
Hence for 10 which has ‘1’ as the non units digit, we need to carry over ‘1’ to the immediate left which turns 8 to 9
1221$^2$ = 1 | 4 | 9 | 0 | 8 | 4 | 1
= 1490841
Example 7: Find the square of 9654
9654$^2$ = D(9) | D(96) | D(965) | D(9654)| D(654) | D(54) | D(4)
= 9$^2$ | 2x9x6 | 2x9x5 + 6$^2$ | 2x9x4+2x6x5 | 2x6x4 + 5$^2$ | 2x5x4 | 4$^2$
= 81 | 108 | 126 | 132 | 73 | 40 | 16
= 93199716