Use the algebraic identity: $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ to evaluate the following: $7.7\times 8.3$

Hint: Try to get values of a and b in the given algebraic equation from the given terms in multiplication. After getting values of a and b put them in the expression $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$. Hence, simplify it further to get the result.

Complete step-by-step answer:
As algebraic identity in the equation is given as
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}................\left( i \right)$
And hence, we need to determine the value of the expression $\left( 7.7 \right)\times \left( 8.3 \right)$.

Now, we can observe the left hand side of the expression and notice that the terms (a – b) and (a +b) are in multiplication i.e. two brackets with same numbers one with difference sign and other has summation sign. It means we need to split 7.7 and 8.3 in two numbers such that the difference of them should be equal to 7.7 and summation of them should be equal to 8.3, because 8.3 is greater than 7.7, so it will go with the summation sign. So, we can observe that 7.7 can be written as (8 – 0.3) and 8.3 can be expressed as (8 + 0.3) which are just similar to the brackets of (a – b) and (a + b) in the given algebraic identity in the problem. It means we can represent 7.7 and 8.3 as (8 – 0.3) and (8 + 0.3) respectively.
Now, the product is,
$7.7\times 8.3=\left( 8-0.3 \right)\left( 8+0.3 \right)................\left( ii \right)$
Now, observe the equation (ii) and equation (i) and hence, on comparing them we get values of a and b as a = 8 and b = 0.3.
Hence, put the value of a and b in the right hand side of the equation (i) as well to get the value of
$7.7\times 8.3$
So, we get
$\left( 8-0.3 \right)\left( 8+0.3 \right)={{\left( 8 \right)}^{2}}-{{\left( 0.3 \right)}^{2}}$
Now, we know the value of ${{8}^{2}}$ is 64 and the value of ${{\left( 0.3 \right)}^{2}}=0.3\times 0.3$ is given as 0.09. So, we get (8 – 0.3) (8 + 0.3) = 64 - 0.09 (or)
(8 – 0.3) (8 + 0.3) = 64.00 - 0.09 = 63.91
So, value of the product of the terms 7.7 and 8.3 is 63.91

Note: We can calculate value of a and b using the following approach:
If given multiplication is xy, where x > y then,
$\begin{align}
  & a=\dfrac{x+y}{2} \\
 & b=\dfrac{x-y}{2} \\
\end{align}$
So, for the given expression
 $\begin{align}
  & 7.7\times 8.3 \\
 & \Rightarrow a=\dfrac{7.7+8.3}{2}=\dfrac{16}{2}=8 \\
 & \Rightarrow \dfrac{8.3-7.7}{2}=\dfrac{0.6}{2}=0.3 \\
\end{align}$
Hence, one may use this approach as well it can be proved with the help of solving equation
a + b = x and a – b = y. One can verify the result by multiplying 7.7 and 8.3 separately as well.