Answer:
The simplest form of given expression [tex]\frac{\sqrt 2-\sqrt 10}{ \sqrt 2+\sqrt 10}[/tex] is [tex]-\frac{3+\sqrt{5}}{2}[/tex]
Step-by-step explanation:
Given expression [tex]\frac{\sqrt 2-\sqrt 10}{ \sqrt 2+\sqrt 10}[/tex]
We are required to simplify the above given expression,
Consider the given expression, [tex]\frac{\sqrt 2-\sqrt 10}{ \sqrt 2+\sqrt 10}[/tex]
We will first rationalize the denominator by multiply and divide by [tex]{\sqrt 2-\sqrt 10}[/tex]
Then , given expression becomes,
[tex]\Rightarrow \frac{\sqrt 2-\sqrt 10}{ \sqrt 2+\sqrt 10}\times \frac{\sqrt 2-\sqrt 10}{\sqrt 2-\sqrt 10}[/tex]
Using identity [tex](a+b)(a-b)=a^2-b^2[/tex] in denominator, we get,
[tex]\Rightarrow \frac{(\sqrt 2-\sqrt 10)( \sqrt 2-\sqrt 10)}{2-10}[/tex]
[tex]\Rightarrow \frac{(\sqrt 2-\sqrt 10)( \sqrt 2-\sqrt 10)}{-8}[/tex]
In numerator, Using identity [tex](a-b)(a-b)=(a-b)^2[/tex], we get,
[tex]\Rightarrow \frac{(\sqrt 2-\sqrt 10)^2}{-8}[/tex]
Using [tex](a-b)^2=a^2+b^2-2ab[/tex] , we get,
[tex]\Rightarrow \frac{ 2+10-4\sqrt{20}}{-8}[/tex]
[tex]\Rightarrow \frac{12+4\sqrt{5}}{-8}[/tex]
On simplifying, we get,
[tex]\frac{12+4\sqrt{5}}{-8}=-\frac{3+\sqrt{5}}{2}[/tex]
Thus, the simplest form of given expression [tex]\frac{\sqrt 2-\sqrt 10}{ \sqrt 2+\sqrt 10}[/tex] is [tex]-\frac{3+\sqrt{5}}{2}[/tex]
