Question

Find the value of ${(82)^2} - {(18)^2}$ .

Answer 1

Hint: In this question we have to find the value of the given expression. So we will try to solve this by using the suitable algebraic identities. We will apply the formula here:
${a^2} - {b^2} = (a + b)(a - b)$ .
By applying the expression from this identity we will apply the formula and then solve it.

Complete step-by-step solution:
Let us first understand the definition of algebraic identity.
We know that algebraic identities are algebraic equations which have variables and constants in them. They are always true for every value of variable in them. We know that algebraic identities have their application in the factorisation of polynomials.
In this question, we will apply the difference of square formula i.e.
${a^2} - {b^2} = (a + b)(a - b)$ .
By comparing from the question, here we have
$a = 82,b = 18$
By applying the formula , we can write
${(82)^2} - {(18)^2} = (82 + 18)(82 - 18)$
On simplifying we have
 $100 \times 64 = 6400$
Hence the required answer is ${(82)^2} - {(18)^2} = 6400$.

Note: We should always remember the algebraic identities and solve them carefully. We know that there are four basic algebraic identities. They are:
a). ${(a + b)^2} = {a^2} + {b^2} + 2ab$
b). ${(a - b)^2} = {a^2} + {b^2} - 2ab$
c). $(x + a)(x + b) = {x^2} + (a + b)x + ab$
d). ${a^2} - {b^2} = (a + b)(a - b)$ .
We have applied the fourth identity in the above solution.
Here we can solve this by using arithmetic operations like first we multiply 82 two times and 18 two times as $81 \times 81 - 18 \times 18= 6561-324$, now on applying subtraction we get 6400.