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Question:

Without calculating, determine if the product
Square rootc75⋅1/3 is rational or irrational.

Answer Options:

It is rational because one factor is rational.

It is rational because both factors are rational.

It is irrational because one of the factors is irrational.

It is irrational because both of the factors are irrational.

1 answer

To determine whether the product \( \sqrt{75} \cdot \frac{1}{3} \) is rational or irrational, we can analyze each factor individually.

  1. Analyzing \( \sqrt{75} \):

    • The number 75 can be factored into \( 25 \times 3 \), where 25 is a perfect square.
    • Therefore, \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \).
    • Since \( \sqrt{3} \) is an irrational number, \( \sqrt{75} \) is also irrational.

  2. Analyzing \( \frac{1}{3} \):

    • \( \frac{1}{3} \) is a rational number because it can be expressed as a fraction of two integers.

Now we can consider the product:

  • We have one factor that is irrational (\( \sqrt{75} \)) and one factor that is rational (\( \frac{1}{3} \)).
  • The product of a rational number and an irrational number is always irrational.

Based on this analysis, the correct answer is:

It is irrational because one of the factors is irrational.