Question

Find the sum of $\sqrt {457.96} $ and $\sqrt {4.5796} $.
A) 2.354
B) 235.4
C) 23.54
D) None of these

Answer 1

Hint:We will first rewrite the given expression by making the numerator to be 45796 and including denominator as required then, from their sum we will take out common of square root of 45796, so that we just have to calculate one square root.

Complete step-by-step answer:
We can clearly see that the numbers we have are similar with just decimal at different places. So, let us just write the expression whose value we actually require:-
We need to find $\sqrt {457.96} + \sqrt {4.5796} $.
We can rewrite it as: $\sqrt {\dfrac{{45796}}{{100}}} + \sqrt {\dfrac{{45796}}{{10000}}} $
Now, we know that $\sqrt {\dfrac{a}{b}} = \dfrac{{\sqrt a }}{{\sqrt b }}$.
Hence, we have our expression as:
$ \Rightarrow \sqrt {45796} \left( {\sqrt {\dfrac{1}{{100}}} + \sqrt {\dfrac{1}{{10000}}} } \right)$ …………..(1)
Now, we know that ${10^2} = 100$ .
Taking reciprocal, we get:-
$ \Rightarrow \dfrac{1}{{{{10}^2}}} = \dfrac{1}{{100}}$
Taking square root of both sides, we will get:-
$ \Rightarrow \sqrt {\dfrac{1}{{{{10}^2}}}} = \sqrt {\dfrac{1}{{100}}} $
Hence, we can write it as:- $ \Rightarrow \sqrt {\dfrac{1}{{100}}} = \dfrac{1}{{10}}$ ………….(2)
Now, we also know that ${100^2} = 10000$ .
Taking reciprocal, we get:-
$ \Rightarrow \dfrac{1}{{{{100}^2}}} = \dfrac{1}{{10000}}$
Taking square root of both sides, we will get:-
$ \Rightarrow \sqrt {\dfrac{1}{{{{100}^2}}}} = \sqrt {\dfrac{1}{{10000}}} $
Hence, we can write it as:- $ \Rightarrow \sqrt {\dfrac{1}{{10000}}} = \dfrac{1}{{100}}$ ………….(3)
Now, putting in (2) and (3) in (1), we will get:-
$ \Rightarrow \sqrt {45796} \left( {\sqrt {\dfrac{1}{{100}}} + \sqrt {\dfrac{1}{{10000}}} } \right) \Rightarrow \sqrt {45796} \left( {\dfrac{1}{{10}} + \dfrac{1}{{100}}} \right)$
Simplifying, we will get:-
$ \Rightarrow \sqrt {45796} \left( {\dfrac{{11}}{{100}}} \right)$ ……….(4)
Now, we just need to calculate the square root of 45796.
We see that:
$ \Rightarrow 45796 = 2 \times 2 \times 107 \times 107$
$ \Rightarrow \sqrt {45796} = \sqrt {2 \times 2 \times 107 \times 107} = 2 \times 107 = 214$
Putting this in (4), we will get:-
$ \Rightarrow \sqrt {457.96} + \sqrt {4.5796} = \dfrac{{11}}{{100}} \times 214$
On simplifying, we will get:-
$ \Rightarrow \sqrt {457.96} + \sqrt {4.5796} = 23.54$

So, the correct answer is “Option C”.

Note: The students must note that if they would have carefully observed the question, they would already have been able to eliminate two of the options that is (A) and (B) because since, we do not have any number with more than 3 digits before decimal, by using Vedic math’s ideology, we can never get a three digit number before decimal in answer as well and similar kind of argument for (B) as well. But you still will have to calculate because the fourth option is none of these and you may get some other number as a result as well.