Question

Find the square root of $7 - 2\sqrt {10} $
$\left( a \right){\text{ }}\sqrt {15} - \sqrt 2 $
$\left( b \right){\text{ }}\sqrt 3 - \sqrt 2 $
$\left( c \right){\text{ }}\sqrt 5 - \sqrt 3 $
$\left( d \right){\text{ }}\sqrt 5 - \sqrt 2 $

Answer 1

Hint:
This question can be done in more than one way. Here we will see the method which needs some basic formula. So first of all we will expand $\sqrt {10} $ and then by applying the formula ${a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}$. By this formula, we will be able to find its square root.

Formula used:
Some basic formula used in this question
${a^2} - 2ab + {b^2} = {\left( {a - b} \right)^2}$
Also, we can write the below as,
$\sqrt {ab} = \sqrt a \sqrt b $
$\sqrt {{a^2}} = \left| a \right|$

Complete step by step solution:
we have to find the $\sqrt {7 - 2\sqrt {10} } $
Now we can write the above expression as
$ \Rightarrow \sqrt {7 - 2\sqrt {5 \cdot 2} } $
As we know $\sqrt {ab} = \sqrt a \sqrt b $
Therefore, on applying the above formula we get
$ \Rightarrow \sqrt {7 - 2\sqrt 5 \sqrt 2 } $
Now on rewriting the$7{\text{ as 5 + 2}}$, we get
$ \Rightarrow \sqrt {5 + 2 - 2\sqrt 5 \sqrt 2 } $
Also, we know that $a + b = b + a$
So from this, we can write the equation as
$ \Rightarrow \sqrt {5 + 2\sqrt 5 \sqrt 2 + 2} $
Now, as we know that ${a^2} - 2ab + {b^2} = {\left( {a - b} \right)^2}$
So on applying the formula and substituting the values, we get
$ \Rightarrow \sqrt {{{\left( {\sqrt 5 - \sqrt 2 } \right)}^2}} $
Therefore, now on applying the rule$\sqrt {{a^2}} = \left| a \right|$, we get
$ \Rightarrow \left| {\sqrt 5 - \sqrt 2 } \right|$
And it can also be written as$\sqrt 5 - \sqrt 2 $.
Therefore, the option $\left( d \right)$ is correct.
Additional information: The square root of a number is a worth that we get when it is increased to itself and produces the first number. For instance, when $5$ is duplicated to itself we get $25$. Accordingly, we can say that 5 is a square root estimation $25$. Similarly, $4$ is the square root estimation of $16$, $6$ is the square root estimation of $36$, and $77$ is the square root estimation of$49$.
Presently, much the same as a square is a portrayal of the region of a square that is equivalent to the side x side, the square root is the portrayal of the length of the side of a square.

Note:
This problem can also be solved by using the formula of${a^2} - 2ab + {b^2} = {\left( {a - b} \right)^2}$. Just by expanding the question, we can have the equation which follows the formula, and then accordingly we will solve them and get the same value. This will be good when you are opting while competitive.