To determine which expressions represent a rational number when \( p \) and \( q \) are rational and \( r \) is an irrational number, consider the following types of operations and combinations:
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Sum of two rational numbers: \( p + q \) is rational.
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Difference of two rational numbers: \( p - q \) is rational.
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Product of two rational numbers: \( p \times q \) is rational.
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Quotient of two rational numbers (provided \( q \neq 0 \)): \( \frac{p}{q} \) is rational.
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Sum of a rational and an irrational number: \( p + r \) is irrational.
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Difference of a rational and an irrational number: \( p - r \) is irrational.
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Product of a rational and an irrational number: \( p \times r \) is irrational (unless \( p = 0 \), in which case it is rational).
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Quotient of a rational and an irrational number: \( \frac{p}{r} \) is irrational (assuming \( r \neq 0 \)).
From these operations, we can conclude that any operation that involves only combining rational numbers (like addition, subtraction, multiplication, or division as long as the divisor is not zero) will yield a rational number.
Thus, any expression solely involving \( p \) and \( q \), such as \( p + q \) or \( p \times q \), will be rational. Other combinations that involve \( r \) (such as \( p + r \)) will yield irrational results.
So, for an expression that represents a rational number, you may use any of the following:
- \( p + q \)
- \( p - q \)
- \( p \times q \)
- \( \frac{p}{q} \) (given that \( q \neq 0 \))
In conclusion, a rational number is represented by expressions solely composed of \( p \) and \( q \).