SOLUTION: True or false (a) If p is rational, then p^2 is rational. (b) If p is irrational, then p^2 is irrational. (c) If p^2 is rational, then p is rational

Question 1209262: True or false
(a) If p is rational, then p^2 is rational.
(b) If p is irrational, then p^2 is irrational.
(c) If p^2 is rational, then p is rational
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

Let m and n represent integers where n is nonzero.
is rational since it's a ratio or fraction of two integers.

is also rational.

and are integers.
Squaring an integer leads to an integer result.
We conclude statement (a) is always true

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Part (b)

This claim is not true for every possible irrational value p. So we must conclude it is false
The claim is true when or for example.

But it's false when . Note that after squaring both sides we get which is rational.

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Part (c)

This statement appears to be true since it's just the reverse of statement (a)

But we can construct this counter-example
--->
which are irrational. We cannot write as a ratio of two integers.

So we conclude the statement is false
Sometimes statement (c) would be true (for instance when ), but as shown above, it's also sometimes false.

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Answers:
(a) true
(b) false
(c) false

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