Question

Rationalise the denominator of the 3+root2/4+root2

Answer 1

Answer:

• $\dfrac{{ 10 + \sqrt{2} } }{{14 }{ } }$

Explanation:

Given fraction,

$\longrightarrow \: \dfrac{ 3+ \sqrt{2} }{4 + \sqrt{2} }$

Multiplying the conjugate of the denominator to the fraction.

$\longrightarrow \: \dfrac{ 3+ \sqrt{2} }{4 + \sqrt{2} } \times \dfrac{4 - \sqrt{2} }{4 - \sqrt{2} }$

$\longrightarrow \: \dfrac{( 3+ \sqrt{2}) {(4 + \sqrt{2} )} }{{(4 - \sqrt{2} )}{(4 - \sqrt{2}) } }$

Applying the identity (a - b)(a + b) = a² - b² on the denominator.

$\longrightarrow \: \dfrac{{ 3(4 - \sqrt{2} ) + \sqrt{2} (4 - \sqrt{2} ) } }{{(4 {)}^{2} -( \sqrt{2} {)}^{2} }{ } }$

$\longrightarrow \: \dfrac{{ 12 - 3\sqrt{2} + 4 \sqrt{2} - {2} } }{{16 -2 }{ } }$

$\longrightarrow \: \dfrac{{ 10 + \sqrt{2} } }{{14 }{ } }$

${ \underline{ \therefore{ \sf{Required \: answer : \: \dfrac{10 + \sqrt{2} }{14} }}}}$

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Identity used :

• (a - b)(a + b) = a² - b²

Conjugate :

• A binomial expression formed by changing the sign. For example conjugate for (x + y) is (x - y)