Question

Show that: 3√11−5 is an irrational number.

Answer 1

Answer:

-5+3√11

Step-by-step explanation:

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Answer 2

Answer:

let us assume that 3√11−5 is rational number .

hence there exists co-prime numbers p and q ( q not equal to 0)

such that ,

$\\ \sf {3\sqrt {11} -5 = \dfrac {p}{q}}$

$\\ \sf {3\sqrt{11} = \dfrac {p}{q} + 5}$

$\\ \sf {3\sqrt{11} = \dfrac {p+5q}{q}}$

$\\ \sf {\sqrt{11}= \dfrac {3p + 5q}{q}}$ -------- (1)

since p and q are integers,

therefore ,

$\\ \sf { \dfrac {3p + 5q}{q} = \dfrac {3\times integer + 5\times integer}{integer (not 0)}}$

$\\ \sf { \dfrac {3p + 5q}{q} = Rational\: number}$

Therefore from (1)

$\\ \sf {\sqrt{11}= \dfrac {3p + 5q}{q} = rational\:number}$

Therefore √11 is rational number

But this contradicts the fact , √11 is irrational number .

Therefore 3√11−5 is an irrational number