Question

4. Rational, irrational (4 points) (1) (2 points) Prove or disprove that if x y is an irrational number, then x or y is also an irrational number. (2) (2 points) Prove that if x 2 is irrational, then x is irrational. (Hint: try a proof by contrapositive)

Answer 1

See explanation below

Explanation:

1) Prove or disprove that if
`x^y` is an irrational number, then x or y is also an irrational number.

Let's take the following instances:

i) When x= 2 and y=
`√(2)` we have:
`2^\sqrt^(^2^)`

ii) When
`x=2√(2)` and y=3, we have:
`(x=2√(2))^3`

iii) When
`x=2√(2)` and
`y = √(2)`, we have:
`(2√(2))^\sqrt^(^2^)`

It is proven because, in scenario

i) x is rational and y is irrational

ii) x is irrational and y is rational

iii) x and y are irrational

2) Prove tha x² is irrational, then x is irrational.

Use contradiction here.

Thus, x² is irrational and x is rational.

`x =(b)/(a)` when x is rational, a & b are integers.

Therefore,
`x^2 =(b^2)/(a^2)`. This x² is rational.

This contradicts the statement that x² is irrational.

Therefore, if x² is irrational, x is also irrational.