Question

If root 5 + root 3 / root 5 - root 3 = a+b root 15 find a and b where a and b are rational number

Answer 1

Given :-

$\sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = a + b \sqrt{15}$

Required to find :-

• Values of " a " and " b "

Identities used :-

• ( x + y ) ( x + y ) = ( x + y )²

• ( x + y ) ( x - y ) = x² - y²

• ( x + y )² = x² + 2xy + y²

Solution :-

Given information :-

$\rm \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } = a + b \sqrt{5}$

we need to find the values of ' a ' and ' b '

So,

Consider the LHS part

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

Here,

We need to rationalize the denominator !

So,

Rationalising factor of √5 - √3 = √5 + √3

Hence,

Multiply both numerator and denominator with that factor

So,

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

Here we need to use some algebraic Identities

They are ,

• 1. ( x + y ) ( x + y ) = ( x + y )²

• 2. ( x + y ) ( x - y ) = x² - y²

• 3. ( x + y )² = x² + 2xy + y²

So,

Using 1 and 2 we get ;

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

Using the 3rd identity expand the numerator

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

This implies,

$\tt \dfrac{5 + 3 + 2 \sqrt{15} }{2}$

$\tt \dfrac{ 8 + 2 \sqrt{15} }{2} \\ \\ \tt \dfrac{2 \: ( \: 4 + \sqrt{15} \: ) }{2}$

2 gets cancelled in both numerator and denominator

So,

we are left with ;

$\tt 4 + \sqrt{15}$

Now,

Compare the LHS and RHS parts

$\rm 4 + \sqrt{15} = a + b \sqrt{15}$

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

So,

Equal the values on both sides

Hence,

a = 4

b = 1

Therefore,

Values of " a " and " b " are 4 & 1

Answer 2

Answer:

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Step-by-step explanation:

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