Question

Find the square of 71 without multiplication.

Answer 1

Hint: In this question, we need to evaluate the value of square of 71 but without actually multiplying the terms. For this, we will bifurcate the number in two parts and use the trigonometric identities to get the result.

Complete step-by-step answer:
When a number is multiplied by itself then, the result obtained is the square of the number. Mathematically, $ x \times x = {x^2} $ where ‘x’ is the number whose square is defined by multiplication of x two times.
The given number is 71 of which we need to evaluate the square.
71 can also be re-written as 71 = (70+1).
According to one of the trigonometric identities, the square of the sum of two numbers is equal to the summation of the square of the individual numbers and twice the product of the numbers. Mathematically, the trigonometric identity is $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $
Comparing the trigonometric identity with the number (70+1) we get, the value of ‘a’ as 70 and the value of ‘b’ as 1.
Substituting the value of a and b in the equation $ {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} $ to determine the square of 71.
  $
\Rightarrow {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} \\
\Rightarrow {\left( {70 + 1} \right)^2} = {(70)^2} + 2(70)(1) + {(1)^2} \\
   = 4900 + 140 + 1 \\
   = 5041 \\
   $
Hence, the square of the number 71 is 5041 which is evaluated without actually multiplying the terms.

Note: Here, the number 71 can also be bifurcated in many ways but we have chosen (70+1) as it is probably the easiest way of squaring the terms. However, students may take any other bifurcation also.