Question

Given that √2 and √3 are irrational, prove that the following are irrational 3 - 5√2​

Answer 1

S O L U T I O N :

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We're given with two irrational numbers √2 and √3. We've to prove that 3 – 5√2 is an irrational number. To do so, we'll assume as that the given number 3 – 5√2 is an rational number. As we know that, rational numbers are the numbers which can be written in the form of ᵐ⁄ₙ, where n ≠ 0.

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Let's say, that 3 – 5√2 is an rational number. So, we can write it in the form of ᵐ⁄ₙ.

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$\longrightarrow\sf 3 - 5\sqrt{2} =\Bigg(\dfrac{m}{n}\Bigg)$

$\longrightarrow\sf -5\sqrt{2} =\Bigg(\dfrac{m}{n}\Bigg) - 3$

$\longrightarrow\sf -5\sqrt{2} = \Bigg(\dfrac{m - 3n}{n}\Bigg)$

$\longrightarrow\sf \sqrt{2} = \Bigg(\dfrac{m - 3n}{-3n}\Bigg)$

$\longrightarrow\sf\sqrt{2} = \Bigg(\dfrac{3n - m}{3n}\Bigg)$

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Here, we can see that 3n – m/3n is an rational number but it is given that √2 is an irrational number and both are equal to each other and this arises the contradiction the fact that √2 is an irrational number.

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∴ Therefore, 3 – 5√2 is an irrational number.

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⠀⠀⠀⠀⠀⠀⠀⠀∴ Hence, Proved!