Question

prove that root 5 + root 11 is irrational

Answer 1

let √5 + √11 be in the form of a/b where a and b are co prime nos and integers

√5 +√11=a/b

√5=a/b-√11

we know that √5 is irrational (as explained below)

therefore √5+√11 is also irrational

hence proved.

Let us assume that √5 is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are integers.

so, √5 = p/q

    p = √5q

we know that 'p' is a rational number. so √5 q must be rational since it equals to p

but it does'nt occurs with √5 since its not an integer

therefore, p =/= √5q

this contradicts the fact that √5 is an irrational number

hence our assumption is wrong and √5 is an irrational number.

Answer 2

Let us consider √5 + √11 is not irrational

then it becomes √5 + √11 is a rational

we know that
p/q where p , q belongs to integers and q ≠ 0


Squaring on both sides

(√5 + √11 ) 2 = ( p/q ) 2

Use ( a + b ) 2 formula

(√5)2 + (√11)2 + 2 × √5 × √11 = p2 / q2
5 + 11 + 2 × √55 = p2 / q2
16 + 2 × √55 = p2 / q2
2 × √55 = p2 / q2 - 16 / 1
√55 = p2 - 16 q2 / q2


LHS = √11 where it is irrational because " 11 " is
not a perfect square .

RHS = p2 - 16 q2 / q2
It becomes rational because it is in the form of p/q


Our contradiction is wrong
Our Assumption is wrong


√5 + √11 is an irrational



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