Question

Prove that root 5 is irrational .hence show that 3root5-8is irratonal

Answer 1

☆Let us assume that √ 5 is a rational number.

√5=p/q.
[where p and q are coprimes and q is not equal to zero. ]

☆Squaring on both sides.
We get,

5=p^2/q^2.

p^2=5q^2...eqn 1.

☆5 divides p^2.

☆Therefore 5 divides p.

☆Thus 5 is a factor of p.

5/p=c.

☆Squaring on both sides we get:

p^2/25=c^2.

p^2=25c^2....eqn.2.

☆From eqn 1 and eqn 2

5q^2=25c^2.

q^2=5c^2.

☆5 divides q^2.

☆Therefore 5 divides q.

☆Thus 5 is a factor of q.

☆5 is a factor of both p and q.

☆But p and q are Co primes.

☆Hence, this contradicts to our assumption.

☆Thus, √ 5 is irrational number.

☆☆Now let us assume that 3√ 5 -8 is rational.

3√5-8=p/q
[where p and q are rational and q is not equal to zero ].

3√5 =p/q+8

3√5=p+8q/q.

√5=p+8q/3q.

☆Here √ 5 is irrational as proved above.

☆Thus irrational is not equal to rational.

☆LHS is not equal to RHS.

☆Thus we can say that 3√5-8 is an irrational number.

Hope it helps. ☆☆