Question

prove that 3root 11 is irrational​

Answer 1

ASSUME THAT 3✓11 IS RATIONAL NUMBER

3√11 = a/b

√11 = a/3b

a/3b is rational. But √5 is irrational . So our assumption is wrong

Hence , 3√11 is irrational

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Answer 2

Let assume root 11 rational
Root 11 = a/b
b root 11= a
Squaring both side
11b square = a square
Therefore a square is divisible by 11
Also a is divisible by 11
11c = a
Squaring both side
121c square = a square
121 c square = 11 b square
(Substituting value)
121 c square/11 = b square
11c square = b square
Therefore b square is divisible by 11
Also b is divisible by 11
This contradict the factor that a and b are co prime
Therefore root 11 is irrational
Let assume 3root 11is rational
3root 11 = a/b
root 11 = a/3b
Therefor a/3b is an integer and root 11 is also integer
But root 11 is irrational
Therefore a/3b is also irrational
Therefore 3root 11 is irrational