Using identities, evaluate the following expression ${\left( {8.9} \right)^2}$.

Hint: Here, we have to evaluate the value of ${\left( {8.9} \right)^2}$ this can be found by using the identity ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$. Write $\left( {8.9} \right)$ as the difference of two numbers whose square can be found quickly or remembered. Then apply the above given identity to find the value of ${\left( {8.9} \right)^2}$.

Complete step-by-step answer:
Here, we have to evaluate the value of ${\left( {8.9} \right)^2}$.
We can write $8.9$ as the difference of $9$ and $0.1$ as the square of $9$ and $0.1$ can be easily calculated or remembered.
Now, we can write ${\left( {8.9} \right)^2} = {\left( {9 - 0.1} \right)^2}$
Here, we can say that $a = 9$ and $b = 0.1$
Now, apply the above given identity ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ to find the required value of ${\left( {8.9} \right)^2}$
$ \Rightarrow {\left( {8.9} \right)^2} = {\left( {9 - 0.1} \right)^2} = {\left( 9 \right)^2} + {\left( {0.1} \right)^2} - 2 \times 9 \times 0.1$
Simplifying this we can write
$ \Rightarrow {\left( {8.9} \right)^2} = 81 + 0.01 - 1.8$
By adding the numbers on the right side we get,
\[ \Rightarrow {\left( {8.9} \right)^2} = 81.01 - 1.8\]
$\therefore {\left( {8.9} \right)^2} = 79.21$

Thus, the required value of ${\left( {8.9} \right)^2}$ is $79.21$

Note: Similarly, this method can be used to evaluate the value of cube of any given number. The given number should be written as the difference or sum of two numbers whose cube is easily calculated or known. Then apply this identity ${\left( {a - b} \right)^3} = {a^3} - 3ab\left( {a - b} \right) - {b^3}$ to evaluate the cube of the given number.
Some other identities which can be used for the evaluation of mathematical expression are
1. ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
2. ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
3. $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$
Identity (1) and (2) are used for the evaluation of the values of similar types of question given above.
Identity (3) is used the evaluation of the mathematical expression like$$ $101 \times 99$
To evaluate the value of this expression we have to break the given two numbers in to two parts one with addition and other with subtraction between the numbers. i.e $\left( {100 + 1} \right)\left( {100 - 1} \right)$
Then apply the identity to evaluate the required value.