Question

how can you write the expression with rationalized denominator 2+sqrt 5/2-sqrt 5

Answer 1

$\dfrac{2+\sqrt5}{2-\sqrt5}=\dfrac{(2+\sqrt5)^2}{4-5}=\dfrac{4+4\sqrt5+5}{-1}=-(9+4\sqrt5)=-9-4\sqrt5$

Answer 2

Answer:

Option D - $-9-4\sqrt5$

Step-by-step explanation:

Given : Expression $\frac{2+\sqrt5}{2-\sqrt5}$

To write : The expression with rationalized denominator.

Solution :

Step 1 - Write the expression

$=\frac{2+\sqrt5}{2-\sqrt5}$

Step 2- Rationalized denominator i.e, multiply and divide the expression with $2+\sqrt5$

$=\frac{2+\sqrt5}{2-\sqrt5}\times \frac{2+\sqrt5}{2+\sqrt5}$

Step 3 - Using identity $(a+b)(a-b)=a^2-b^2$ in the denominator.

$=\frac{(2+\sqrt5)^2}{2^2-(\sqrt5)^2}$

Step 4 - Solve the expression

$=\frac{2^2+(\sqrt5)^2+2(2)(\sqrt5)}{4-5}$

$=\frac{4+5+4\sqrt5}{-1}$

$=-9-4\sqrt5$

The solution is rewrite as  $\frac{2+\sqrt5}{2-\sqrt5}=-9-4\sqrt5$

Therefore, Option D is correct.