Question

5+3root2/5-3root2 = a+b root2.

Find the value of A and B

Answer 1

Explanation:

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Answer 2

Required to find :-

• Values of " a " and " b "

Method used :-

• Rationalising the denominator

Identities used :-

1. ( x + y ) ( x + y ) = ( x + y )²

2. ( x + y ) ( x - y ) = x² - y²

3. ( x + y )² = x² + y² + 2xy

Solution :-

$\tt{ \dfrac{ 5 + 3 \sqrt{2} }{ 5 - 3 \sqrt{2} } = a + b \sqrt{2} }$

We need to find the values of a and b

So,

Consider the LHS part

$\rightarrow{\rm{ \dfrac{ 5 + 3 \sqrt{2} }{ 5 - 3 \sqrt{2}} }}$

$\boxed{\tt{ Rationalising \; factor \; of \; 5 - 3 \sqrt{3} = 5 + 3 \sqrt{3}}}$

$\rightarrow{\rm{ \dfrac{ 5 + 3 \sqrt{2}}{ 5 - 3 \sqrt{2} } \times \dfrac{ 5 + 3 \sqrt{2}}{ 5 + 3 \sqrt{2}}}}$

Using the identities

1. ( x + y ) ( x + y ) = ( x + y )²

2. ( x + y ) ( x - y ) = x² - y²

3. ( x + y )² = x² + y² + 2xy

So,

By using 1st and 2nd identities . we get ;

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

By using the 3rd identity expand the numerator

$\rightarrow{\rm{ \dfrac{ {(5)}^{2} + {(3 \sqrt{2} )}^{2} + 2 ( 5 ) ( 3 \sqrt{2}) }{ 25 - 18 }}}$

$\rightarrow{\rm{ \dfrac{ 25 + 18 + 30\sqrt{2} }{ 7 }}}$

$\rightarrow{\rm{ \dfrac{ 43 + 30\sqrt{2} }{ 7 }}}$

Now split the numerator along with the denominator

$\rightarrow{\rm{ \dfrac{43}{7} + \dfrac{30\sqrt{2}}{7} }}$

$\rightarrow{\rm{ \dfrac{43}{7} + \dfrac{30 \sqrt{2} }{7} }}$

Hence,

Compare the LHS and RHS

$\implies{\tt{ \dfrac{43}{7} + \dfrac{30 \sqrt{2}}{7} = a + b\sqrt{2} }}$

Since,

LHS is in the form of RHS

Equal the values on both sides

Hence ,

a = 43/7

b = 30/7

$\large{\boxed{\rm{\therefore{Values \; of \; ' a ' \; and \; ' b ' = \dfrac{43}{7} \; and \; \dfrac{30}{7}}}}}$