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The number ww is rational. Which statement about 22+w\sqrt{22} + w is true?\newlineChoices:\newline(A) 22+w\sqrt{22} + w is rational.\newline(B) 22+w\sqrt{22} + w is irrational.\newline(C) 22+w\sqrt{22} + w can be rational or irrational, depending on the value of ww.

Full solution

Q. The number ww is rational. Which statement about 22+w\sqrt{22} + w is true?\newlineChoices:\newline(A) 22+w\sqrt{22} + w is rational.\newline(B) 22+w\sqrt{22} + w is irrational.\newline(C) 22+w\sqrt{22} + w can be rational or irrational, depending on the value of ww.
  1. Identify Type of Number: Identify whether 22\sqrt{22} is a rational or irrational number.2222 is a non-perfect square.22\sqrt{22} is an irrational number.
  2. Nature of Sum: Consider the nature of the sum of a rational number ww and an irrational number 22\sqrt{22}. The sum of a rational number and an irrational number is always irrational.
  3. Truth of Statements: Determine the truth of the statements given the nature of 22\sqrt{22} and ww. Since ww is rational and 22\sqrt{22} is irrational, their sum 22+w\sqrt{22} + w is irrational. Therefore, the correct statement is that 22+w\sqrt{22} + w is irrational.

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