Question

Find the square of 86 without multiplication.

Answer 1

Hint: Here, we will express 86 as a sum of two numbers such that one of the numbers is a multiple of 10. Then we will square the sum and simplify it using the suitable algebraic identity to get the required answer

Complete step-by-step answer:
Firstly we will split the number in $a + b$ form where $a$ should be a multiple of $10$.
So 86 can be written as
$86 = 80 + 6$
Squaring both the sides, we get
$ \Rightarrow {\left( {86} \right)^2} = {\left( {80 + 6} \right)^2}$
Using formula ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$, we get
$ \Rightarrow {(86)^2} = {(80)^2} + {(6)^2} + 2 \times 80 \times 6$
Applying the exponent the terms, we get
$ \Rightarrow {(86)^2} = 6400 + 36 + 960$
Adding the terms, we get
$ \Rightarrow {(86)^2} = 7396$
So square of 86 is 7396.

Note: We can use the diagonal method also to find the square of the number.
First we will make a square and divide it into 4 parts as we have 2 digits. We will multiply the two digits on the left of the row and top most of the column. We will then write the number such that the digit at 10th place is above the diagonal and the digit at unit place is below the diagonal of the sub-square between the two digits.

	$8$	$6$
$8$	$6$ $4$	$4$ $8$
$6$	$4$ $8$	$3$ $6$

Now we will get the square of the number from backward by each diagonal of the square. Starting from the lower most diagonal we will write the digit. If there is more than one digit we will add them and then right the unit place digit first and add the 10th place digit to the next diagonal as follows:
Digit at the lower most diagonal is 6 so the last digit of square is 6.
Next, the 2nd most diagonal has three digits and on adding them we get
$8 + 3 + 8 = 19$
So, the next digit after 6 is 9 and 1 will be added to the next diagonal which has 3 digits. Therefore, we get
$4 + 4 + 4 + 1 = 13$
So the next digit after 6,9 is 3 and 1 will be added to the next diagonal which has 1 digit. Therefore, we get
$6 + 1 = 7$
So we get our answer as
${(86)^2} = 7396$