Question

prove that √3+√7 is an irrational number​

Answer 1

Step-by-step explanation:

Question : Prove that√3 + √7 is irrational.

Answer :

Let us assume that √3 + √7 is a rational number.

So it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

⇒√3 + √7 =p/q

On squaring both the sides we get,

⇒(√3 + √7)² =p²/q²

⇒3 + 7 + 2√21 = p²/q²

⇒10 + 2√21q²=p² —————–(i)

p²/10 + 2√21= q²

So 10 + 2√21 divides p

p is a multiple of 10 + 2√21

⇒p=10 + 2√21m

⇒p = (10 + 2√21)²m²

⇒p = (100 + 84 + 2(10)(2√21))m²

⇒p²= 184 + 40√21m² ————-(ii)

From equations (i) and (ii), we get,

10 + 2√21 q²= 184 + 40√21m²

⇒q²= 10 + 2√21m²

⇒q² is a multiple of 10 + 2√21

⇒q is a multiple of 10 + 2√21

Hence, p,q have a common factor 10 + 2√21. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√3 + √7 is an irrational number

Hence proved