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The number bb is irrational. Which statement about 33+b\sqrt{33} + b is true?\newlineChoices:\newline(A)33+b\sqrt{33} + b is rational.\newline(B)33+b\sqrt{33} + b is irrational.\newline(C)33+b\sqrt{33} + b can be rational or irrational, depending on the value of bb.

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Q. The number bb is irrational. Which statement about 33+b\sqrt{33} + b is true?\newlineChoices:\newline(A)33+b\sqrt{33} + b is rational.\newline(B)33+b\sqrt{33} + b is irrational.\newline(C)33+b\sqrt{33} + b can be rational or irrational, depending on the value of bb.
  1. Identify Number Type: Identify whether 33\sqrt{33} is a rational or irrational number.3333 is a non-perfect square.33\sqrt{33} is an irrational number.
  2. Consider Sum of Irrational Numbers: Consider the sum of two irrational numbers.\newlineWe know:\newlinebb is an irrational number.\newline33\sqrt{33} is an irrational number.\newlineThe sum of two irrational numbers is not necessarily irrational. For example, if b=33b = -\sqrt{33}, then 33+b=3333=0\sqrt{33} + b = \sqrt{33} - \sqrt{33} = 0, which is rational.
  3. Determine Nature of Sum: Determine the possible nature of 33+b\sqrt{33} + b. If bb is any irrational number other than 33-\sqrt{33}, then 33+b\sqrt{33} + b will be irrational because the sum of an irrational number and a rational number is irrational. However, if b=33b = -\sqrt{33}, then 33+b\sqrt{33} + b is rational. Therefore, 33+b\sqrt{33} + b can be rational or irrational, depending on the value of bb.

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