Question

Simplify the following expressions:
$(\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 )$

Answer 1

Hint: In this question we will use the concept of the operations on real numbers . Here we will use some identities relating to the square roots. We will follow the distributive law of multiplication over addition of real numbers by using the identity $(x + y)(x - y) = {x^2} - {y^2}$, for any real numbers $x$and $y$.

Complete step-by-step solution -
Here we have,
 $(\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 )$ ………(i)
By using the identity , we know that
$(x + y)(x - y) = {x^2} - {y^2}$ ………(ii)
Now, using equation (ii), we have
$x = \sqrt 5 ,y = \sqrt 2 $
Therefore , using these values in equation (ii), we get
$
   \Rightarrow (\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 ) = {(\sqrt 5 )^2} - {(\sqrt 2 )^2} \\
   \Rightarrow (\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 ) = 5 - 2 \\
   \Rightarrow (\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 ) = 3 \\
$
Hence , we can say that $(\sqrt 5 - \sqrt 2 )(\sqrt 5 + \sqrt 2 ) = 3$ .

Note: In this type of questions , we will always use the identities of real numbers .First we have to identify which identity is correct by comparing the given equation with the identity and then by putting the suitable values in the identity we will get the required answer. Here the identity we have used is $(x + y)(x - y) = {x^2} - {y^2}$ where $x$ and $y$ are real numbers.