Question

Examine whether the following is rational or irrational:

(√2 + √3)^2

Answer 1

Answer:

Step-by-step explanation:

We know that (a+b)^2 = a^2 + b^2 + 2ab

therefore

(√2 + √3)^2 = 2 + 3 + 2(√2)(√3) = 5+2√6

now, we know that any rational number added to an irrational number, always yields an irrational number.

to prove

lets consider 5 + 2√6 to be rational

let a = 5 + 2√6

a-5 = 2√6

(a-5)/2 = √6

now, according to our assumption a is rational and we know that 5 and 2 are also rational, therefore, a-5 is rational and (a-5)/2 in turn is a fraction which is also a rational.

but √6 is not rational! hence our assumption is wrong, and therefore   5+2√6 is irrational = (√2 + √3)^2 is irrational

hope it helps, CHEERS MATE,

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