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The number qq is irrational. Which statement about q+18q + \sqrt{18} is true?\newlineChoices:\newline(A)q+18q + \sqrt{18} is rational.\newline(B)q+18q + \sqrt{18} is irrational.\newline(C)q+18q + \sqrt{18} can be rational or irrational, depending on the value of qq.

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Full solution

Q. The number qq is irrational. Which statement about q+18q + \sqrt{18} is true?\newlineChoices:\newline(A)q+18q + \sqrt{18} is rational.\newline(B)q+18q + \sqrt{18} is irrational.\newline(C)q+18q + \sqrt{18} can be rational or irrational, depending on the value of qq.
  1. Given Information: We are given that qq is an irrational number. The square root of 1818, 18\sqrt{18}, can be simplified to 9×2\sqrt{9\times2} which is 3×23\times\sqrt{2}. Since 2\sqrt{2} is an irrational number, 3×23\times\sqrt{2} is also irrational. The sum of an irrational number and another irrational number is always irrational.
  2. Simplifying 18\sqrt{18}: Therefore, q+18q + \sqrt{18} which is q+32q + 3\sqrt{2} is the sum of two irrational numbers. By the property mentioned earlier, the sum of two irrational numbers is irrational.
  3. Sum of Irrational Numbers: The correct statement about q+18q + \sqrt{18} is that it is irrational, which corresponds to choice (B)(B).

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