Question

Using identities, evaluate: ${99^2}$.

Answer 1

Hint: Here we go through by writing the term 99 as (100-1) because in the question we have to evaluate using identities. So for solving this always think of the two numbers whose square we find easily. And apply the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$.

Complete step-by-step answer:
Here we write 99 as (100-1)
And the identity we use is ${(a - b)^2} = {a^2} - 2ab + {b^2}$
We have to evaluate our number using this identity.
So we can say (100-1) is the same as (a-b) so we simply put a as 100 and b as 1 in the given identity to evaluate.
By putting the values we get,
${\left( {100 - 1} \right)^2} = {100^2} - 2 \times 100 \times 1 + {1^2}$
Now simplify the term that is on the right hand side,
i.e. R.H.S= ${100^2} - 2 \times 100 \times 1 + {1^2}$
=10000-200+1
=9801

Hence we get the answer by the help of identities.
For cross checking you simply find the square of 99 i.e. $99 \times 99 = 9801$ which also gives the same answer.

Note: Whenever we face such a type of question the key concept for solving the question is always think of the number in that way whose square you easily find. Here in this question the simple way is (100-1) you can also choose different numbers such as (102-3) for the identities, but in this case for finding the square of 102 is little bit complex as finding the square of 100. So always choose a simple number to prove the identities.