Question

Prove that under root 8+5 is Irrational

Answer 1

Answer: proof is given below.

Step-by-step explanation:

suppose √8 = a/b with integers a, b

and gcd(a,b) = 1 (meaning the ratio is simplified)

then 8 = a²/b²

and 8b² = a²

this implies 8 divides a² which also means 8 divides a.

so there exists a p within the integers such that:

a = 8p

and thus,

√8 = 8p/b

which implies

8 = 64p²/b²

which is:

1/8 = p²/b²

or:

b²/p² = 8

which implies

b² = 8p²

which implies 8 divides b² which means 8 divides b.

8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1

therefore, the square root of 8 is irrational.

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Answer 2

Answer:

prove. that 8 root 5 is irrational