Question

rationalise 1 /√2 + 1​

Answer 1

Step-by-step explanation:

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Answer 2

To rationalise : 1 / (√2 + 1 )

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Solution :

In order to rationalise denominator of any fraction, we have to multiply both numerator and denominator with the conjugate of the denominator.

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Conjugate of ( √2 + 1 ) which is the denominator of the fraction is (√2 - 1 ), this means we have to multiply both numerator and denominator with ( √2 - 1 ).

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$\dashrightarrow \tt \dfrac{1}{( \sqrt{2} + 1) } \times \dfrac{( \sqrt{2} - 1 )}{( \sqrt{2} - 1)}$

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$\dashrightarrow \tt \dfrac{( \sqrt{2} - 1 )}{( \sqrt{2} + 1)( \sqrt{2} - 1)}$

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Now we can see that the denominator is of the form (A + B) (A - B) = A² - B² for A = √2 and B =1.

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$\dashrightarrow \tt \dfrac{( \sqrt{2} - 1 )}{( \sqrt{2} ) ^{2} - (1)^{2} }$

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$\dashrightarrow \tt \dfrac{( \sqrt{2} - 1 )}{( 2 - 1) }$

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$\dashrightarrow \tt \dfrac{( \sqrt{2} - 1 )}{( 1) }$

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$\dashrightarrow \tt ( \sqrt{2} - 1 )$

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$\dashrightarrow \tt \sqrt{2} - 1$

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Hence 1 / ( √2 + 1 ) is rationalised to ( √2 - 1 ).