Question

Suppose x and y are different irrational numbers. Mark each statement as ALWAYS,

SOMETIMES, or NEVER true by using the drop down at the end of each statement.

3x is an irrational number. [Select]

X^2 is a rational number. [Select]

X • Y is a rational number. [Select]

X + 3 is an irrational number. [Select]

X + Y is a rational number. [Select]

X - Y is an irrational number.

Select]

Answer 1

always True

sometimes true

sometimes true

always true

sometimes true

always true

Explanation:

As product of non-zero rational number and irrational number is irrational,

3x is an irrational number : always True

Take
`x=\sqrt[3]{2}`

Here,
`x` is an irrational number.

`x^2=(\sqrt[3]{2})^2=2^{(2)/(3) }` is also an irrational number

Now take
`x=√(2)`

Here,
`x` is an irrational number.

`x^2=(√(2))^2=2` is a rational number

So,

`x^2` is a rational number: sometimes true

Take
`x=√(2) \,,y=√(3)`

Here,
`x,y` are irrational numbers.

`xy=√(2)√(3)=√(6)` is also an irrational number.

Now take
`x=√(2) \,,y=√(8)`

Here,
`x,y` are irrational numbers.

`xy=√(2) √(8)=√(16)=4` is a rational number.

So,

`xy` is a rational number: sometimes true

As sum of a rational number and an irrational number is always irrational,

`x+3` is an irrational number: always true

Take
`x=√(2) \,,y=-√(2)`

Here,
`x,y` are irrational numbers.

`x+y=√(2) +(-√(2))=0` is a rational number

Now take
`x=√(2)\,,\,y=√(3)`

Here,
`x,y` are irrational numbers.

`x+y=√(2)+√(3)` is an irrational number.

So,

`x+y` is a rational number: sometimes true

As difference of two irrational numbers is always irrational,

`x-y` is an irrational number: always true