Question

If both a and b are rational numbers, find the values of a and b:
√3 -1
V3+1
= a +b V3​

Answer 1

Correct Question :-

If both a and b are rational numbers, find the values of a and b:

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$\sf \dfrac{\sqrt{3}-1}{\sqrt{3}+1} = a + b\sqrt{3}$

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Answer :-

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➩ $\sf \dfrac{\sqrt{3}-1}{\sqrt{3}+1} = a + b\sqrt{3}$

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Rationalizing the denominator -

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➩ $\sf \Big(\dfrac{\sqrt{3}-1}{\sqrt{3}+1}\Big) \times \Big(\dfrac{\sqrt{3}-1}{\sqrt{3}-1}\Big)$

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➩ $\sf \dfrac{\Big(\sqrt{3}-1\Big)\Big(\sqrt{3}-1\Big)}{\Big(\sqrt{3}-1\Big)\Big(\sqrt{3}+1\Big)}$

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➩ $\sf \dfrac{(\sqrt{3})^2 + (1)^2 - 2 \times ( \sqrt{3} ) (1) }{(\sqrt{3})^2 - (1)^2}$

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➩ $\sf \dfrac{3+1-2\sqrt{3}}{3-1}$

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➩ $\sf \dfrac{4-2\sqrt{3}}{2}$

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➩ $\sf 2 - \sqrt{3}$

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Comparing LHS and RHS :-

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➩ $\sf 2 - \sqrt{3} = a + b\sqrt{3}$

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$\boxed{\sf a = 2 }$

$\boxed{\sf b = -1}$