Evaluate the given expression \[{\left( {8.63} \right)^2} - {\left( {1.37} \right)^2}\].

Hint: Here, the given expression is a difference of squares of two numbers. We will use the algebraic identity for the difference of the squares of two numbers and simplify the expression to find the required value.

Formula Used:
The difference of the squares of two numbers can be calculated using the algebraic identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\].

Complete step-by-step answer:
The expression \[{\left( {8.63} \right)^2} - {\left( {1.37} \right)^2}\] is a difference of squares of two numbers \[8.63\] and \[1.37\].
The difference of the squares of two numbers can be calculated using the algebraic identity \[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\].
Substituting \[a = 8.63\] and \[b = 1.37\] in the algebraic identity, we get
\[ \Rightarrow {\left( {8.63} \right)^2} - {\left( {1.37} \right)^2} = \left( {8.63 - 1.37} \right)\left( {8.63 + 1.37} \right)\]
Adding and subtracting the terms in the parentheses, we get
\[ \Rightarrow {\left( {8.63} \right)^2} - {\left( {1.37} \right)^2} = \left( {7.26} \right)\left( {10} \right)\]
Multiplying \[7.26\] by 10 in the expression, we get
\[ \Rightarrow {\left( {8.63} \right)^2} - {\left( {1.37} \right)^2} = 72.6\]
\[\therefore \] The value of the difference of squares \[{\left( {8.63} \right)^2} - {\left( {1.37} \right)^2}\] is \[72.6\].

Note: We multiplied \[7.26\] by 10 in the solution. The product of a decimal number by a power of 10 (10, 100, 1000, etc) can be calculated using a simple method. The number of zeroes in 10 is 1. When a decimal number is multiplied by 10, the result can be obtained by shifting the decimal point 1 digit to the right. For example: When \[7.26\] is multiplied by 10, the result is \[72.6\], which is simply obtained by shifting the decimal point one place to the right from before 2 to before 6.
We can verify the solution by finding the square of \[8.63\] and \[1.37\], and subtracting them.
The square of \[8.63\] is \[74.4769\].
The square of \[1.37\] is \[1.8769\].
The difference of the squares of \[8.63\] and \[1.37\] \[ = 74.4769 - 1.8769 = 72.6\]
Hence, we have verified that the value of the difference of squares \[{\left( {8.63} \right)^2} - {\left( {1.37} \right)^2}\] is \[72.6\].