Suppose that p and q and r is an irrational number which expression represents a rational number?

A. P + r
B. P + pr
C. P + q
D. P + qr

1 answer

In order to identify which expression represents a rational number, we need to keep in mind that the sum or product of a rational number and an irrational number is irrational, while the sum or difference of two rational numbers is rational.

Let's analyze each option:

A. \( p + r \)

  • Since \( p \) is rational and \( r \) is irrational, \( p + r \) is irrational.

B. \( p + pr \)

  • Here, \( pr \) (the product of \( p \) which is rational and \( r \) which is irrational) is irrational. Adding \( p \) (rational) to \( pr \) (irrational) gives us an irrational number.

C. \( p + q \)

  • Both \( p \) and \( q \) are rational, so \( p + q \) is rational.

D. \( p + qr \)

  • The term \( qr \) involves \( q \) (which is rational) and \( r \) (which is irrational), making \( qr \) irrational. Therefore, \( p + qr \) (rational + irrational) is also irrational.

The only expression that represents a rational number is:

C. \( p + q \).