Question

let x and y be ratiinal and irrational numbers respectively .Are x+y and xy irriational ? justify your answer by an example of each

Answer 1

hiii!!!

here's ur answer...

$let \: be \:x\: 3 \: and\:y\:be\: \sqrt{5}$

$\sqrt{5}=2.23606798...$ it's an irratinal number as it's non-terminating and non-repeating.

$therefore\: x + y= 3 + \sqrt{5} \\ = 3 + 2.23606798... \\ = 5.23606798...$

hence, it's necessary that rational + irrational is always a irrational number.

now, come to xy...

like x + y = irrational, xy = is also irrational

but sometimes it can be rational too..

for example, we know 0 is a rational number as it can be represented in the form of p/q and q not equal to 0

let x be 0 and y be under root 5

xy = 0 * under root 5

= 0

we know that the product with 0 is always 0.

so here, it's not necessary that rational * irrational is always irrational.

hope this helps..!!

Answer 2

Hola!
Thanks for the question :)

In the 1st case,
Let
x = ✓7
And y = 4

We know,
x + y = ✓7 + 4 =2.64575.. + 4
=> x + y = 6.64575

This will continue till infinity.
=> x + y will be irrational.

Now,
Let
x = ✓7
y = 4

xy = 7×✓4 = 7✓4
=> xy = 10.583...

This number again continues till infinity.
=> xy will be irrational.

Exception :
Let's say we have two irrational numbers
x = ✓2 and y = 0
So,
xy = ✓2 × 0 = 0
Here,
0 is a rational number.

This says that xy can also be rational.

Peace out ✌️