The correct response to prove that a rational number multiplied by an irrational number gives an irrational product is:
by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.
Here’s a brief outline of how this proof can be structured:
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Definition of Rational and Irrational Numbers: Recall that a rational number can be expressed as the fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). An irrational number cannot be expressed as a fraction of integers.
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Assume a Rational Number and an Irrational Number: Let \( r = \frac{a}{b} \) be a rational number and let \( x \) be an irrational number.
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Multiply the Rational Number by the Irrational Number: Consider the product \( r \times x = \frac{a}{b} \times x = \frac{ax}{b} \).
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Prove by Contradiction: Now, assume for the sake of contradiction that the product \( r \times x \) is rational. This would mean that \( ax \) must be rational since \( b \) is non-zero and rational.
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If \( ax \) is rational and \( a \) (a non-zero integer) is rational, then \( x \) must also be rational (because the product of a non-zero rational number and an irrational number cannot be rational). This is a contradiction because \( x \) was assumed to be irrational.
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Conclusion: Therefore, our initial assumption that \( r \times x \) is rational must be false, which means that the product \( r \times x \) is irrational.
This proof successfully demonstrates that multiplying a rational number by an irrational number leads to an irrational number.