Question

$\sqrt{11} - \sqrt{7} \div \sqrt{11} + \sqrt{7} = a - b \sqrt{77}$
then find the value of a and b



please answer this question by step by step explanation ​

Answer 1

hope it will help you.....

Answer 2

Required to find :-

• Values of 'a' and 'b'

Solution :-

Given :-

$\rm \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } = a - b \sqrt{77}$

Consider the LHS part ;

$\bf \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} }$

Here,

We need to rationalise the denominator

Rationalising factor of √11 + √7 = √11 - √7

Multiply both numerator and denominator with rationalising factor

So,

$\bf{ \rm \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } \times \dfrac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} - \sqrt{7} } }$

Here we need to use some algebraic identities

The Identities are ;

• 1. ( x - y ) ( x - y ) = ( x - y )²

• 2. ( x + y ) ( x - y ) = x² - y²

• 3. ( x - y )² = x² + y² - 2xy

Now,

Using the 1st and 2nd identity

$\bf \rm \dfrac{ \big( \sqrt{11} - \sqrt{7} \: \: {\big )}^{2} }{( \sqrt{11} {)}^{2} - ( \sqrt{7} {)}^{2} }$

Expand the numerator using the 3rd identity

So,

$\bf \rm \dfrac{( \sqrt{11} {)}^{2} + ( \sqrt{7} {)}^{2} - 2( \sqrt{11})( \sqrt{7} ) }{11 - 7}$

Solving further ;

$\bf \rm \dfrac{11 + 7 - 2 \sqrt{77} }{4}$

$\bf \rm \dfrac{ 18 - 2 \sqrt{77} }{4}$

$\bf \rm \dfrac{2 \: (9 - 1 \sqrt{77} \: \: )}{4}$

Cancelling 2 in numerator and 4 in denominator leaving 2

$\bf \rm \dfrac{9 - 1 \sqrt{77} }{2}$

By splitting the above one into 2 parts

$\bf \rm \: \dfrac{9}{2} - \dfrac{1 \sqrt{77} }{2}$

Now consider both LHS and RHS part

$\bf \rm \dfrac{9}{2} - \dfrac{1 - \sqrt{77} }{2} = a - b \sqrt{77}$

From the above consideration we can conclude that the LHS is in the form of RHS

So, Equal the parts on both sides

Hence, We get ;

>> Value of 'a' = 9/2

>> Value of 'b' = 1/2