Which statement describes the product of a nonzero rational number and an irrational number?

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Prepare for the Praxis Middle School Mathematics Exam with quizzes. Study with flashcards and multiple choice questions. Each question provides hints and detailed explanations. Ensure success on your test!

Multiple Choice

Which statement describes the product of a nonzero rational number and an irrational number?

Explanation:
When you multiply a nonzero rational number by an irrational number, the result must be irrational. Here’s why: let r be a nonzero rational and i be irrational, and suppose their product ri is rational. Then you could solve for i as i = (ri)/r. Since ri is assumed rational and r is a nonzero rational, their quotient would be rational, which would make i rational—contradicting that i is irrational. Therefore the product cannot be rational and must be irrational. It can’t be zero either, because r is nonzero and i is not zero, so their product can’t vanish. So the description that fits is an irrational product.

When you multiply a nonzero rational number by an irrational number, the result must be irrational. Here’s why: let r be a nonzero rational and i be irrational, and suppose their product ri is rational. Then you could solve for i as i = (ri)/r. Since ri is assumed rational and r is a nonzero rational, their quotient would be rational, which would make i rational—contradicting that i is irrational. Therefore the product cannot be rational and must be irrational. It can’t be zero either, because r is nonzero and i is not zero, so their product can’t vanish. So the description that fits is an irrational product.

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