Question

2. Rationalize thedenominater
of 1/4+2√3​

Answer 1

Given :

$:\normalsize\boxed{\bf\dfrac{1}{4+2\sqrt{3}}}$

To Find :

Rationalise the denominator.

Solution :

Analysis :

Here we have to rationalise by using the required formula and the evaluation signs.

Explanation :

$\\ :\implies\normalsize\sf\dfrac{1}{4+2\sqrt{3}}$

Rationalizing the denominator,

$\\ :\implies\normalsize\sf\dfrac{1}{4+2\sqrt{3}}\times\dfrac{4-2\sqrt{3}}{4-2\sqrt{3}}$

Now multiplying,

$\\ :\implies\normalsize\sf\dfrac{1(4-2\sqrt{3})}{(4+2\sqrt{3})\times(4-2\sqrt{3})}$

Using the formula (a + b)(a - b) = (a² - b²) in the denominator,

$\\ :\implies\normalsize\sf\dfrac{4-2\sqrt{3}}{(4)^2-(2\sqrt{3})^2}$

Removing the squares in denominator,

$\\ :\implies\normalsize\sf\dfrac{4-2\sqrt{3}}{16-(4\times3)}$

After evaluation,

$\\ :\implies\normalsize\sf\dfrac{4-2\sqrt{3}}{16-12}$

$\\ :\implies\normalsize\sf\dfrac{4-2\sqrt{3}}{4}$

Taking 2 as common in numerator,

$\\ :\implies\normalsize\sf\dfrac{2(2-\sqrt{3})}{4}$

$\\ :\implies\normalsize\sf\dfrac{\cancel{2}(2-\sqrt{3})}{\cancel{4}}$

After evaluation,

$\\ :\implies\normalsize\sf\dfrac{2-\sqrt{3}}{2}$

$\\ \normalsize\therefore\boxed{\bf\dfrac{2-\sqrt{3}}{2}}$

The answer is (2 - √3)/2.