Simplify: $\sqrt{42+\sqrt{49}}$

Hint: This question mainly involves the concept of solving the square root of a positive integer. We all know about the concept of finding the square root of a positive number.

Complete step-by-step answer:
We know that if there is a positive integer, which is the square of a number, let us say ‘a’. Therefore, the answer to that square root will be that number only.

Let us see this approach:

If $\sqrt{n}=\sqrt{a\times a}$

Here, n is any positive integer and it is also the square of the integer a.

Also, we know that to find the square root of a number, we have to raise that number to the

power half i.e. $\dfrac{1}{2}$

$\begin{align}

  & n=a\times a={{a}^{2}} \\

 & \Rightarrow \sqrt{n}={{n}^{\dfrac{1}{2}}}={{({{a}^{2}})}^{\dfrac{1}{2}}}=a \\

\end{align}$ As power to the number and its whole power gets multiplied.

 So, to solve the above question, step by step we have to solve the given square roots which will ultimately result in our answer.


Hence, to solve the above problem, the first step involved is to solve $\sqrt{49}$

As 49 is the square of 7 i.e. $49={{7}^{2}}$

$\Rightarrow \sqrt{49}=7$ ……..(i)

Now, we will proceed with the further steps involved.

$\begin{align}

  & \therefore \sqrt{42+\sqrt{49}}=\sqrt{42+7} \\

 & \Rightarrow \sqrt{42+\sqrt{49}}=\sqrt{49}=7 \\

\end{align}$

Now, with the help of (i), we know that $\sqrt{49}=7$ .

Hence, the answer to the above question is 7.

Note: These types of questions are very simple. But, we have to analyse whether the given number is a perfect square or not. Also, we have to find that number whose square is this number. In some cases, like when we have to find the square root of a non-perfect square number or a decimal number, then we have to undergo large calculations which increases the chances of committing a mistake.