Question

Evaluate using identity: ${108^2}$.

Answer 1

Hint: In the given question, we have to evaluate a square of a number given to us in the problem itself with the help of an algebraic identity. The algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is used to evaluate the square of a binomial expression involving the sum of two terms.

Complete step by step answer:
Given question requires us to find the value of a square of $108$. We can use the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ for the task described. So, to calculate the square of $108$, we have to first divide the number $108$ into two parts such that the following calculation of the square of the number becomes easier.

So, we know that $108 = 100 + 8$. So, we can divide $108$ into the two numbers $100$ and $8$. So, we now substitute the values of the two parts into the algebraic identity that we are supposed to use. Hence, we have,
${108^2} = {\left( {100 + 8} \right)^2}$
Now, we expand the left side of the equation using the algebraic identity to evaluate the square of a binomial expression involving the sum of two terms. So, we get,
${108^2} = {\left( {100} \right)^2} + 2\left( {100} \right)\left( 8 \right) + {\left( 8 \right)^2}$
$ \Rightarrow {108^2} = 10000 + 1600 + 64$.
Simplifying the expression further, we get,
$ \therefore {108^2} = 11664$

So, the value of ${108^2}$ calculated using the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is $11664$.

Note:Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. We can also verify the answer of the given question by calculating the square of $108$ simply.