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The number ww is irrational. Which statement about w−5w - \sqrt{5} is true?\newlineChoices:\newline(A) w−5w - \sqrt{5} is rational.\newline(B) w−5w - \sqrt{5} is irrational.\newline(C) w−5w - \sqrt{5} can be rational or irrational, depending on the value of ww.

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Q. The number ww is irrational. Which statement about w−5w - \sqrt{5} is true?\newlineChoices:\newline(A) w−5w - \sqrt{5} is rational.\newline(B) w−5w - \sqrt{5} is irrational.\newline(C) w−5w - \sqrt{5} can be rational or irrational, depending on the value of ww.
  1. Identify Type of Number: Identify whether 5\sqrt{5} is a rational or irrational number. 55 is a non-perfect square, which means that its square root cannot be expressed as a ratio of two integers. Therefore, 5\sqrt{5} is an irrational number.
  2. Properties of Irrational Numbers: Consider the properties of irrational numbers. The difference between two irrational numbers can be either rational or irrational. It depends on the specific numbers involved. For example: - If w=5w = \sqrt{5}, then w−5=5−5=0w - \sqrt{5} = \sqrt{5} - \sqrt{5} = 0, which is rational. - If ww is any irrational number not related to 5\sqrt{5}, then w−5w - \sqrt{5} is likely to be irrational because the difference of two unrelated irrational numbers is typically irrational. Therefore, without additional information about ww, we cannot definitively say whether w−5w - \sqrt{5} is rational or irrational.

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