Question

Prove that root 5 is an irrational number hence show that 2+root 5 is also an irrational

Answer 1

Answer:

Step-by-step explanation:

Let's prove this by the method of contradiction-

Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers.

⇒√5=p/q

⇒5=p²/q²  {Squaring both the sides}

⇒5q²=p²  (1)

⇒p² is a multiple of 5. {Euclid's Division Lemma}

⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}

⇒p=5m

⇒p²=25m²   (2)

From equations (1) and (2), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5. {Euclid's Division Lemma}

⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}

Hence, p,q have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.  

For you second query, as we've proved √5 irrational. Therefore √5+3 is also irrational because sum of a rational and an irrational number is always an irrational number.

Answer 2

Answer:

click these pics for answer

Step-by-step explanation:

its a prove that root 5 is irrational and 5+4 root 5 is also an irrational no.