State whether this is true or false.The is irrational, where p, q are primes.

# State whether this is true or false.The is irrational, where p, q are primes.

1. A
True
2. B
False

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### Solution:

Here, we'll assume that and  is rational, where p and q are distinct primes,
Thus, . Where, x is rational.
Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides.
Thus, x, x², p, q, & 2 are all rational, and rational numbers are closed under subtraction and division.
So, is rational.
But since p and q are both primes, then pq is not a perfect square and therefore ​is not rational. But this is a contradiction. Original assumption must be wrong.
So, and  is rational, where p and q are distinct primes.
Hence, the given statement is true.

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