Question

Find the sum of \[\sqrt {457 \cdot 96} \] and \[\sqrt {4 \cdot 5796} \]
A. 2.354
B. 235.4
C. 23.54
D. None of these

Answer 1

Hint: Convert a decimal number into fraction by writing the number in numerator (without decimal point) and writing \[{10^n}\] in denominator where \[n\] is the number of place where the decimal is placed (counting from right end). Write the terms in numerator and denominator as square of a number so as to cancel the square root. Add the two terms in the end.
* Prime factorization is a process of writing a number in multiple of its factors where all factors are prime numbers.

Complete step-by-step answer:
We have to find the sum of \[\sqrt {457 \cdot 96} \] and \[\sqrt {4 \cdot 5796} \]
We calculate the square root of each number separately.
(i)\[\sqrt {457 \cdot 96} \]
We first write the number under the square root in form of fraction
Since number of places after which decimal is placed from the right end is 2
So, the numerator is 45796 and denominator is 100
The fraction becomes \[\dfrac{{45796}}{{100}}\]
\[ \Rightarrow \sqrt {457 \cdot 96} = \sqrt {\dfrac{{45796}}{{100}}} \]................… (1)
Write prime factorization of numerator and denominator
\[ \Rightarrow 45796 = 2 \times 2 \times 107 \times 107\]
Since base is same we can add powers
\[ \Rightarrow 45796 = {2^2} \times {107^2}\]
Since powers are same we can multiply the bases
\[ \Rightarrow 45796 = {(2 \times 107)^2}\]
\[ \Rightarrow 45796 = {(214)^2}\]..............… (2)
\[ \Rightarrow 100 = 5 \times 5 \times 2 \times 2\]
Since base is same we can add powers
\[ \Rightarrow 100 = {5^2} \times {2^2}\]
Since powers are same we can multiply the bases
\[ \Rightarrow 100 = {(2 \times 5)^2}\]
\[ \Rightarrow 100 = {(10)^2}\]...............… (3)
Substitute values from prime factorization from equations (2) and (3) in fraction in equation (1)
\[ \Rightarrow \sqrt {457 \cdot 96} = \sqrt {\dfrac{{{{214}^2}}}{{{{10}^2}}}} \]
Collect the base in under root in RHs of the equation
\[ \Rightarrow \sqrt {457 \cdot 96} = \sqrt {{{\left( {\dfrac{{214}}{{10}}} \right)}^2}} \]
Cancel square root by square power
\[ \Rightarrow \sqrt {457 \cdot 96} = \dfrac{{214}}{{10}}\]...............… (4)
(ii)\[\sqrt {4 \cdot 5796} \]
We first write the number under the square root in form of fraction
Since number of places after which decimal is placed from the right end is 4
So, the numerator is 45796 and denominator is 10000
The fraction becomes \[\dfrac{{45796}}{{10000}}\]
\[ \Rightarrow \sqrt {4 \cdot 5796} = \sqrt {\dfrac{{45796}}{{10000}}} \]..............… (5)
Write prime factorization of numerator and denominator
Numerator:
\[ \Rightarrow 45796 = 2 \times 2 \times 107 \times 107\]
Since base is same we can add powers
\[ \Rightarrow 45796 = {2^2} \times {107^2}\]
Since powers are same we can multiply the bases
\[ \Rightarrow 45796 = {(2 \times 107)^2}\]
\[ \Rightarrow 45796 = {(214)^2}\]............… (6)
Denominator:
\[ \Rightarrow 10000 = 5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2\]
Since base is same we can add powers
\[ \Rightarrow 10000 = {5^4} \times {2^4}\]
Since powers are same we can multiply the bases
\[ \Rightarrow 10000 = {(2 \times 5)^4}\]
\[ \Rightarrow 10000 = {(100)^2}\].................… (7)
Substitute values from prime factorization from equations (6) and (7) in fraction in equation (5)
\[ \Rightarrow \sqrt {4 \cdot 5796} = \sqrt {\dfrac{{{{214}^2}}}{{{{100}^2}}}} \]
Collect the base in under root in RHs of the equation
\[ \Rightarrow \sqrt {4 \cdot 5796} = \sqrt {{{\left( {\dfrac{{214}}{{100}}} \right)}^2}} \]
Cancel square root by square power
\[ \Rightarrow \sqrt {4 \cdot 5796} = \dfrac{{214}}{{100}}\]...................… (8)
Now add equations (4) and (8)
\[ \Rightarrow \sqrt {457 \cdot 96} + \sqrt {4 \cdot 5796} = \dfrac{{214}}{{10}} + \dfrac{{214}}{{100}}\]
Take LCM in RHS of the equation
\[ \Rightarrow \sqrt {457 \cdot 96} + \sqrt {4 \cdot 5796} = \dfrac{{2140 + 214}}{{100}}\]
\[ \Rightarrow \sqrt {457 \cdot 96} + \sqrt {4 \cdot 5796} = \dfrac{{2354}}{{100}}\]
Place the decimal in RHS at two places from the right side to convert fraction to decimal
\[ \Rightarrow \sqrt {457 \cdot 96} + \sqrt {4 \cdot 5796} = 23 \cdot 54\]
\[\therefore \]The sum of \[\sqrt {457 \cdot 96} \] and \[\sqrt {4 \cdot 5796} \] is $23.54$

\[\therefore \]Option C is correct.

Note: Number under the square root must never be zero. Number under square can be zero. Students are likely to make mistakes while calculating the value under the square root when they attempt to directly write the value of the root without converting into fraction form. Fraction form helps us to clearly place the decimal point in the final answer.