Question

Find the square root of the expression, $11+2\sqrt{30}$.

Answer 1

Hint: In order to find the solution of this question, we will consider the square root of $11+2\sqrt{30}$ as $a+\sqrt{b}$ because the square root of an irrational number is always irrational. Also, we should know that ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$. By using this we will compare the root of $11+2\sqrt{30}$ by it and we will get the answer.

Complete step-by-step answer:
In this question, we have been asked to find the square root of $11+2\sqrt{30}$. To solve this question, we will consider $a+\sqrt{b}$ as the square root of $11+2\sqrt{30}$ because the square root of an irrational number is always irrational. So, we can write,
${{\left( a+\sqrt{b} \right)}^{2}}=11+2\sqrt{30}$
Now, we know that ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$. So, for x = a and $y=\sqrt{b}$, we can say, ${{\left( a+\sqrt{b} \right)}^{2}}={{a}^{2}}+b+2a\sqrt{b}$. Therefore, we can write,
${{a}^{2}}+b+2a\sqrt{b}=11+2\sqrt{30}$
Now, we will compare the rational and irrational part of both sides of the equation. So, we get,
${{a}^{2}}+b=11$ and $2a\sqrt{b}=2\sqrt{30}$
$\Rightarrow {{a}^{2}}+b=11.........\left( i \right)$ and $a=\dfrac{\sqrt{30}}{\sqrt{b}}.........\left( ii \right)$
From equation (ii), we will put the value of a in equation (i). So, we get,
$\begin{align}
  & {{\left( \dfrac{\sqrt{30}}{\sqrt{b}} \right)}^{2}}+b=11 \\
 & \dfrac{30}{b}+b=11 \\
 & \dfrac{30+{{b}^{2}}}{b}=11 \\
 & 30+{{b}^{2}}=11b \\
 & {{b}^{2}}-11b+30=0 \\
\end{align}$
And, by middle term spitting method, we can write,
$\begin{align}
  & {{b}^{2}}-5b-6b+30=0 \\
 & b\left( b-5 \right)-6\left( b-5 \right)=0 \\
 & \left( b-5 \right)\left( b-6 \right)=0 \\
\end{align}$
$b-5=0$ or $b-6=0$
b = 5 or b = 6
Now, we will put both the values of b one by one in equation (ii). So, we get,
For b = 5, $a=\dfrac{\sqrt{30}}{\sqrt{5}}=\sqrt{6}$ and for b = 6, $a=\dfrac{\sqrt{30}}{\sqrt{6}}=\sqrt{5}$.
Therefore, for b = 5 and $a=\sqrt{6}$, we get $\left( a+\sqrt{b} \right)=\left( \sqrt{6}+\sqrt{5} \right)$ and, for b = 6 and $a=\sqrt{5}$, we get $\left( a+\sqrt{b} \right)=\left( \sqrt{5}+\sqrt{6} \right)$. In both the cases we got \[\left( a+\sqrt{b} \right)=\left( \sqrt{5}+\sqrt{6} \right)\]. Hence, we can say that the square root of $11+2\sqrt{30}$ is \[\left( \sqrt{5}+\sqrt{6} \right)\].

Note: We can also solve this question by hit and trial method, by writing $\sqrt{30}=\sqrt{2}\times \sqrt{15}=\sqrt{6}\times \sqrt{5}=1\times \sqrt{30}$ and then by choosing the correct possible pair, which might give us the correct answer for this small value. But if we get some bigger value and then apply the hit and trial method, it will waste a lot of time. So, it is better to solve this question using the conventional method.