Question

The square root of $11 - \sqrt {120} $ is given by

Answer 1

Hint: For solving this question we will first assume the $11 - \sqrt {120} $ as $\sqrt p - \sqrt q $ and then we will square it and so we have the two-equation and from this, we will find the value of $p\& q$ and so we will get the values by putting in it.

Formula used:
Algebraic formula,
${\left( {a - b} \right)^2} = {a^2} + {b^2} + 2ab$
Here, $a\& b$ are the variables.

Complete step-by-step answer:
On taking the prime factors of $120$ , we have
\[2 \times 2 \times 2 \times 3 \times 5\]
So, it can be written as
$ \Rightarrow 120 = 2\sqrt {30} $
So the question $11 - \sqrt {120} $ can be written as $11 - 2\sqrt {30} $
Let us assume the square root of $11 - 2\sqrt {30} $ being $\sqrt p - \sqrt q $
Now we will square it, we get
$ \Rightarrow p - 2\sqrt {pq} + q = 11 - 2\sqrt {30} $
And on comparing the equation, we get
$ \Rightarrow p + q = 11$ , we will name it equation $1$
And $pq = 30$ , and we will name it equation $2$
From equation $1$
$ \Rightarrow p = 11 - q$
On substituting the values, we get
$ \Rightarrow \left( {11 - q} \right)q = 30$
And on solving the multiplication, we get
$ \Rightarrow 11q - {q^2} = 30$
Now taking all the terms to the right side and also taking the negative sign common, so we get
$ \Rightarrow {q^2} - 11q + 30 = 0$
Now on factoring the above equation, we get
$ \Rightarrow (q - 6)(q - 5) = 0$
Therefore, on solving it, we get
$ \Rightarrow q = 6{\text{ or }}q = 5$
Therefore from the value of $p$ will be equal to
$ \Rightarrow p = 11 - 6$
And on solving the subtraction we get
$ \Rightarrow p = 5$
Or, $p = 11 - 5$
And on solving the subtraction we get
$ \Rightarrow p = 6$
Hence, substituting the values in $\sqrt p - \sqrt q $ , we get
$ \Rightarrow \sqrt 5 - \sqrt 6 {\text{ or }}\sqrt 6 - \sqrt 5 $
Therefore, the value for the square root of $11 - \sqrt {120} $ will be $\sqrt 5 - \sqrt 6 {\text{ or }}\sqrt 6 - \sqrt 5 $ .

Note: There is a various method to calculate the square root like prime factorization or long division method and many more. Here for easy understanding and easy to solve we had used prime factorization and some algebraic concepts to solve this and we can see how easily we solved it.