Question

prove that
$\sqrt{3 } - \sqrt{7}$
is am irrational numbers

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Answer 1

Answer:

Step-by-step explanation:

Assuming √3–√7 is rational

Then

a/b=√3–√7 ----------(1)  (a,b are integer)

Then b/a =1/√3–√7

 b/a= 4/√3–

=4(√3+√7) / (√3–√7)(√3+√7)

=(√3+√7) / 3-7

b/a=-1/4(√3+√7)

4b/a= - √3 - √7 ---------------(2)

adding (1) and (2)

a/b +4b/a=√3–√7 -√3–√7 = -2√7

a/b +4b/a=-2√7

Here a/b is rational  so b/a is rational

so a/b +4b/a is rational

hence left side is rational

so -2√7 is rational

or √7 is rational

but we know that √7 is irrational

hence our assumption is not correct

hence √3–√7 is irrational