Question

prove that 5-√3 is irrational given that √3 is irrational

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Answer 1

let us assume that 5-√3 I a rational number

5-√3=a/b,where a,b are coprimes and b=0

5-√3 =a/b

5=a/b+√3

5= a+√3b/b

L.H.S =5 is rational number,R.H.S=a+√3b/b is irrational number

a rational number is never equals to an irrational number so our assumption is wrong

and 5-√3 I an irrational number

Answer 2

Step-by-step explanation:

Let us suppose the opposite that 5-√3 is rational and can be writren in the form of a/b where b is not equal to 0 and both a and b are integers.

So, 5-√3 = a/b

=> -√3 = a/b -5

=> -√3 = (a-5b)/b

=> √3 = -{ (a-5b)/b }

=> √3 = (5b-a)/b

Since, a,b and 5 are rational

Therefore, (5b-a)/b is also rational

Then, by this √3 is also rational but it is given that √3 is irrational.

Since, Rational is not equal to irrational.

So, our assumption that 5-√3 is rational is wrong.

Hence proved that 5-√3 is irrational.

Hope you're satisfied with my answer

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