Question

1) x and y be rational and irtational numbers , respectively. Is x+ y necessarily a rational number? Give an example in support of your answer.<br /><br />2) x be rational and y be irrational. Is xy necessarily irrational ? justify your answer and give an example.<br /><br />

Answer 1

The sum of a rational and an irrational number is always irrational as the non perfect square root will no get cancelled. For example,
$x = 3 \\ y = 2 + \sqrt{3} \\ x + y = 5 + 2 \sqrt{3}$
Here in the rational number that is x, there is no surd to cancel the surd in the irrational number.

The product of such numbers is also always irrational as to make the √3 to 3, it must be multiplied by √3 itself but it is an irrational number!!