Column Method
This method uses the identity (a + b)^{2} = a^{2} + 2ab + b^{2} Step 1 : Find 57^{2} Here a = 5 and b =7
Column I
Column II
Column III
a^{2}
2 x a x b
b^{2}
5^{2}= 25
2 x 5 x 7 =70
7^{2}= 49
Step II: Underline the digit of b^{2}( in column III) and add its tens digit, if any, to 2 x a x b (in column III)
Column I
Column II
Column III
a^{2}
2 x a x b
b^{2}
25
70 + 4 = 74
49
Step III: Underline the digit in column II and add the number formed by tens and other digit, if any, to a^{2} in column I.
Column I
Column II
Column III
a^{2}
2 x a x b
b^{2}
25 + 7
70 + 4 = 74
49
Step IV: under the number in column I
Column I
Column II
Column III
a^{2}
2 x a x b
b^{2}
32
74
49
Write the underlined digits from the unit digit.
Therefore, 57^{2} = 3,249 .
_________________________________________________________________ Examples
1) Find the squares of the following numbers using column method: (i) 99 (ii) 89 Solution:(i) Here, a = 9 and b = 9.
We have,
Column I
Column II
Column III
a^{2}
2 x a x b
b^{2}
9^{2}
2 x 9 x 9
9^{2}
81
162
81
81 + 17
162 + 8 = 170
81
98
170
81
Therefore, 99^{2} = 9801
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(ii) 89^{2}
Here, a = 8, b = 9.
We have,