Question

Prove that 4 + 3√11 is an irrational number.​

Answer 1

Let 4 + 3√11 is rational number

Therefore, it can be written as;

4+ 3√11 = p/q

after cross-multiplication;

(4+3√11)q = p

on squaring both sides;

[(4+3√11)q]² = p²

(16 + 3 ×11) q² = p²

(16+33)q² = p²

49q² = p² → (i)

p² is divisible by 49 ie, p is also divisible by 49.

p/49 = r (r is the result obtained)

49r = p

on squaring both sides;

(49r)² = p²

2401 r² = p² → (ii)

on comparing eqⁿ (i) and (ii), we get;

49q² = 2401r²

q² = 49r²

q² is divisible by 49 ie, q is also divisible by 49.

hence, p and q have same factor.

p and q in rational number do not have common factor

therefore, 4+ 3√11 is not rational number

So, 4+ 3√11 is an irrational number