Question

prove that 3 root 5 minus 8 is irrational number

Answer 1

Let us assume on the contrary that 3√5-8 is a rational number ,then there exists a and b co-prime integers such that,
3√5-8=a/b

3√5=a/b+8

3√5=(a+8b)/b

√5=(a+8b)/3b

(a+8b)/3b is a rational number.
Then, √5 is also a rational number.
But we know that √5 is an irrational number.
This is a contradiction.
This contradiction arisen as our assumption is wrong.
Hence 3√5-8 is an irrational number.

Answer 2

Let us assume that 3√5 - 8 is a rational number of the form a/b where a and b are coprime integers.
3√5 - 8 = a/b
√5 - 8= a/b *3
√5 = 3a/8b
=> √5 is rational number.
But this contradicts the fact that √5 is irrational.
This contradiction has arisen due to our incorrect assumption.
Therefore 3√5 - 8 is an irrational number.