Question

Find the square root of $\sqrt{13-4\sqrt{3}}$.

Answer 1

Hint: For solving this question, we should know about square root and the properties of square root for each type of question and for the given equation. The square root is the under root of any digit which is a perfectly squared digit or not and it is always defined for positive value digits. For the negative values it gives imaginary values.

Complete step by step answer:
In the question it is given that we have to find the square root of $\sqrt{13-4\sqrt{3}}$. For the calculation of square root in this question, we have to divide it in two parts, as follows,
Let $\sqrt{13-4\sqrt{3}}=\sqrt{x}-\sqrt{y}$
Now take whole square on both sides, so we get,
$\begin{align}
  & 13-4\sqrt{3}=x+y-2\sqrt{xy} \\
 & \Rightarrow 13-\left( 4\sqrt{3} \right)=x+y-\left( 2\sqrt{xy} \right)\ldots \ldots \ldots \left( 1 \right) \\
\end{align}$
With the help of equation (1), we can find the values of $x+y$ and $xy$. So,
$\Rightarrow x+y=13$ and $xy=12$
Now, the square of $x+y$ is,
$\begin{align}
  & {{\left( x+y \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}} \\
 & \Rightarrow {{\left( 13 \right)}^{2}}=\left( {{x}^{2}}+{{y}^{2}} \right)+24 \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}=169-24 \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}=145 \\
\end{align}$
Now, we have to calculate the values of ${{\left( x-y \right)}^{2}}$. So, we get,
$\begin{align}
  & {{\left( x-y \right)}^{2}}=\left( {{x}^{2}}-2xy+{{y}^{2}} \right) \\
 & \Rightarrow {{\left( x-y \right)}^{2}}=145-24 \\
 & \Rightarrow {{\left( x-y \right)}^{2}}=121 \\
\end{align}$
Now, take the square root for getting the values of $x-y$, so,
$\Rightarrow x-y=11$
So, we have found the value of $x=12,y=1$, therefore,
$\sqrt{13-4\sqrt{3}}=\sqrt{12}-1\ldots \ldots \ldots \left( 2 \right)$
Now, we have to find the square root of this equation number (2). So, the square root will be,
$\sqrt{13-4\sqrt{3}}=\sqrt{\sqrt{12}-1}$

Hence, we have calculated the square root of $\sqrt{13-4\sqrt{3}}$ and it is $\sqrt{\sqrt{12}-1}$.

Note: While solving these types of questions, you should take the square root carefully for the values which are not completely square values or whose roots will not be an integer value. And you have to take roots carefully for all the values.