Question

Prove that root 6 + root 2 is irrational

Answer 1

since p and q are integers p2 -4q2/2pq is a rational and so √2 is also rational

but this contradicts the fact that √2 is rational

this is the contradiction to the fact that √6+√2 is not rational

thus it is proved that √6+√2 is irrational

Answer 2

√6 + √2 is irrational is proved

Given :

The number √6 + √2

To find :

To prove √6 + √2 is irrational

Solution :

Step 1 of 2 :

Write down the given number

The given number is √6 + √2

Step 2 of 2 :

Prove that √6 + √2 is irrational

We shall prove by method of contradiction

Let us assume that

√6 + √2 is rational

⇒ (√6 + √2)² is rational

⇒ ( 6 + 2√12 + 2 ) is rational

⇒ ( 8 + 4√3 ) is rational

⇒ ( 8 + 4√3 ) - 8 is rational [ Subtraction of two rational number is rational ]

⇒ 4√3 is rational

⇒ √3 is rational

Which is a contradiction as √3 is irrational

So our assumption was wrong

Hence √6 + √2 is irrational

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