Using a suitable identity evaluate \[{(105)^2}\].

Hint: We have evaluated a suitable identity for the required term. We know that an algebraic identity is an equality that holds for any values of its variables. At first, we check the nearest hundreds of 105. We will try to write 105 in terms of the nearest hundred. Since, \[105\] is close to 100. So, we can write it as \[105 = 100 + 5\]. Then, we can apply the algebraic formula and substitute the values into that we will get the final answer.

Formula used: \[{(a + b)^2} = {a^2} + 2ab + {b^2}\]

Complete step-by-step answer:
It is given that; \[{(105)^2}\]
We have to evaluate the value of \[{(105)^2}\] by using suitable identity. With the help of the identities we can get any value quickly.
An algebraic identity is an equality that holds for any values of its variables.
\[105\] is close to 100. So, we can write it as \[105 = 100 + 5\]
Now, we will apply the identity of
\[{(a + b)^2} = {a^2} + 2ab + {b^2}\]
Here, \[a = 100,b = 5\]
Substitute the in the above identity we have,
$\Rightarrow$\[{(105)^2} = {100^2} + 2 \times 100 \times 5 + {5^2}\]
Simplifying we get,
$\Rightarrow$\[{(105)^2} = 10000 + 1000 + 25\]
Simplifying again we get,
$\Rightarrow$\[{(105)^2} = 11025\]

Hence, \[{(105)^2} = 11025\]

Note: Algebraic identities in maths refer to an equation that is always true regardless of the values assigned to the variables.
Identity means that the left-hand side of the equation is identically equal to the right-hand side, for all values of the variables.
Since an identity holds for all values of its variables, it is possible to substitute instances of one side of the equality with the other side of the equality.
For example, because of the identity above, we can replace any instance \[{(a + b)^2}\] with \[{a^2} + 2ab + {b^2}\] and vice versa.
For the above sum, we can justify the answer by multiplying 105 again with 105. We get the same answer.
$ \Rightarrow 105 \times 105 = 11025$
Hence we got.