Find the value of \[{\left( {4.9} \right)^2}\]

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Hint: A square of a number is the product of number by itself represented by\[{\left( n \right)^2}\], where n is the number whose square is to be found and\[{\left( {} \right)^2}\]is the operator of the square.
In this question, we will use some arithmetic identities for finding the square of a decimal number. The first step is to convert the number into \[\left( {a - b} \right)\]or into \[\left( {a + b} \right)\]form and then, the numbers are squared. By using the identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.


Complete step by step solution:
\[{\left( {4.9} \right)^2}\]is the number whose square is to be found, but it can be seen that number is in decimal form so we will have it into \[\left( {a - b} \right)\]or \[\left( {a + b} \right)\],
We know
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2} - - - - (i)\]
\[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2} - - - - (ii)\]
Hence we can write \[{\left( {4.9} \right)^2}\] as \[{\left( {5 - 0.1} \right)^2}\]
By using the formula\[(ii)\] where \[a = 5\]and \[b = 0.1\]
\[
  {\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2} \\
  {\left( {5 - 0.1} \right)^2} = {\left( 5 \right)^2} - 2\left( 5 \right)\left( {0.1} \right) + {\left( {0.1} \right)^2} \\
   = 25 - \left( {10} \right)\left( {0.1} \right) + 0.01 \\
   = 25 - 1 + 0.01 \\
   = 24.01 \\
 \]
Hence square of the number\[{\left( {4.9} \right)^2} = 24.01\]


Additional Information: For the product of the decimal number we multiply the number without considering the decimal, now count the number of digits after the decimal place in both the number and place the decimal point in the result counting the total number of digit from the right side equal to the sum of the decimal place.
For the number\[{\left( {0.1} \right)^2} = 0.1 \times 0.1\] find the product of the number without considering the decimal i.e. \[1 \times 1 = 1\], now count the digits after the decimal for both numbers which is 2, now place the decimal point in the result by counting the digits from the right side\[ = 0.01\].




Note: Alternatively, the square of the number is found by multiplying the number by itself but in the case of a decimal number, we convert the decimal number into fraction number\[\dfrac{a}{b}\], where the numerator and the denominator are squared separately.