Question

Q.28: Rationalise the denominators of the following:

(i)1/√7

(ii)1/(√5+√2)

(iii)5/(√3-√5)​

Answer 1

Given :

• 1/√7
• 1/(√5+√2)
• 5/(√3-√5)

To Find :

• Rationalise the denominators .

Solution :

1. 1 / √7 :

$\longmapsto\tt{\dfrac{1}{\sqrt{7}}\times\dfrac{\sqrt{7}}{\sqrt{7}}}$

$\longmapsto\tt\bf{\dfrac{\sqrt{7}}{7}}$

_______________________

2. 1 / (√5+√2) :

$\longmapsto\tt{\dfrac{1}{\sqrt{5}-\sqrt{2}}\times\dfrac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}}$

$\longmapsto\tt{\dfrac{\sqrt{5}-\sqrt{2}}{(\sqrt{5}+\sqrt{2)}\:(\sqrt{5}-\sqrt{2)}}}$

Using Identity : (a+b) (a-b) = a² - b²

$\longmapsto\tt{\dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{(5)}^{2}-\sqrt{(2)}^{2}}}$

$\longmapsto\tt{\dfrac{\sqrt{5}-\sqrt{2}}{5-2}}$

$\longmapsto\tt\bf{\dfrac{\sqrt{5}-\sqrt{2}}{3}}$

_______________________

3. 5 / √3 - √5 :

$\longmapsto\tt{\dfrac{5}{\sqrt{3}-\sqrt{5}}\times\dfrac{\sqrt{3}+\sqrt{5}}{\sqrt{3}+\sqrt{5}}}$

Using Identity : (a+b) (a-b) = a² - b²

$\longmapsto\tt{\dfrac{5\sqrt{3}+\sqrt{5)}}{\sqrt{(3)}^{2}-\sqrt{(5)}^{2}}}$

$\longmapsto\tt{\dfrac{5\sqrt{(3}+\sqrt{5)}}{3-5}}$

$\longmapsto\tt\bf{\dfrac{5\sqrt{(3}+\sqrt{5)}}{-2}}$

$\longmapsto\tt\bf{\dfrac{-5\sqrt{(3}+\sqrt{5)}}{2}}$

Answer 2

$\Large \dag\underline{\underline{ \green{\sf Given:}}}\dag$

✭ $\red{\sf \dfrac{1}{\sqrt{7}}}$

✭ $\red{\sf \dfrac{1}{\sqrt{5}+\sqrt{2}}}$

✭ $\red{\sf \dfrac{5}{\sqrt{3}-\sqrt{5}}}$

$\Large \dag\underline{\underline{ \green{\sf Find:}}}\dag$

✬ $\purple{\sf Rationalise \: the \: denominator}$

$\Large \dag\underline{\underline{ \green{\sf Solution:}}}\dag$

➊

$\to \pink{\sf \dfrac{1}{ \sqrt{7} } }$

$\to \pink{\sf \dfrac{1}{ \sqrt{7}} \times \dfrac{ \sqrt{7} }{ \sqrt{7} } }$

$\to \pink{\sf \dfrac{ \sqrt{7} }{ {(\sqrt{7})}^{2} } }$

$\to \pink{\sf \dfrac{ \sqrt{7} }{7} }$

____________________

➋

$\to \blue{\sf \dfrac{1}{ \sqrt{5} + \sqrt{2} } }$

$\to \blue{\sf \dfrac{1}{ \sqrt{5} + \sqrt{2} } \times \dfrac{ \sqrt{5} - \sqrt{2} }{ \sqrt{5} - \sqrt{2} } }$

$\to\blue{\sf \dfrac{ \sqrt{5} - \sqrt{2} }{ {(\sqrt{5})}^{2} - {(\sqrt{2})}^{2} } } \\ \bigg [ \sf (a + b)(a - b) = {a}^{2} - {b}^{2} \bigg ]$

$\to\blue{\sf \dfrac{ \sqrt{5} - \sqrt{2} }{ 5 - 2} }$

$\to\blue{\sf \dfrac{ \sqrt{5} - \sqrt{2} }{ 3} }$

____________________

➌

$\to\orange{\sf \dfrac{5}{\sqrt{3}-\sqrt{5}}}$

$\to\orange{\sf \dfrac{5}{\sqrt{3}-\sqrt{5}} \times \dfrac{ \sqrt{3} + \sqrt{5}}{ \sqrt{3} + \sqrt{5}} }$

$\to\orange{\sf \dfrac{ 5\sqrt{3} + 5\sqrt{5} }{ {(\sqrt{3})}^{2} - {(\sqrt{5})}^{2} } } \\ \bigg [ \sf (a + b)(a - b) = {a}^{2} - {b}^{2} \bigg ]$

$\to\orange{\sf \dfrac{ 5\sqrt{3} + 5\sqrt{5} }{ 3 - 5} }$

$\to\orange{\sf \dfrac{ 5\sqrt{3} + 5\sqrt{5} }{ - 2} }$

$\to\orange{\sf \dfrac{ 5(\sqrt{3} + \sqrt{5})}{ - 2} }$

$\to\orange{\sf \dfrac{ - 5(\sqrt{3} + \sqrt{5})}{2} }$