Question

Prove (5+3√2) irrational where √2 is irrational

Answer 1

Any whole number added, subtracted, multiplied or divided from an irrational number, the answer comes out to be irrational.


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Answer 2

Answer:

5+3√2

Let us assume, to the contrary, that 5+3√2 is rational, such that it can be expressed in the form a/b, where a and b are integers and co-prime.

5+3√2= $\dfrac{a}{b}$

⇒ 3√2 = $\dfrac{a}{b}$ - 5

⇒ 3√2 = $\dfrac{a-5b}{b}$

⇒ √2 = $\dfrac{a-5b}{3b}$

Since, a and b are integers, therefore $\dfrac{a-5b}{3b}$  must be rational. So, √2 is also rational.

But this contradicts the fact that √2 is irrational.

Therefore, 5+3√2 is irrational.