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The number tt is irrational. Which statement about 32t\sqrt{32} - t is true?\newlineChoices:\newline(A) 32t\sqrt{32} - t is rational.\newline(B) 32t\sqrt{32} - t is irrational.\newline(C) 32t\sqrt{32} - t can be rational or irrational, depending on the value of tt.

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Q. The number tt is irrational. Which statement about 32t\sqrt{32} - t is true?\newlineChoices:\newline(A) 32t\sqrt{32} - t is rational.\newline(B) 32t\sqrt{32} - t is irrational.\newline(C) 32t\sqrt{32} - t can be rational or irrational, depending on the value of tt.
  1. Identify Type of Number: Identify whether 32\sqrt{32} is a rational or irrational number. 3232 is a perfect square of 44 times 88, which means 32=4×8=4×8=2×8\sqrt{32} = \sqrt{4 \times 8} = \sqrt{4} \times \sqrt{8} = 2 \times \sqrt{8}. Since 8\sqrt{8} is not a perfect square, it is irrational. Therefore, 32\sqrt{32} is irrational.
  2. Properties of Irrational Numbers: Consider the properties of irrational numbers. The difference between two irrational numbers can be rational or irrational. For example, if t=32t = \sqrt{32}, then 32t=0\sqrt{32} - t = 0, which is rational. However, if tt is any other irrational number, then 32t\sqrt{32} - t is likely to be irrational.
  3. Determine Always True Statement: Determine the statement that is always true.\newlineSince 32t\sqrt{32} - t can be rational if t=32t = \sqrt{32}, but is irrational for any other irrational tt, the correct statement is that 32t\sqrt{32} - t can be rational or irrational, depending on the value of tt.

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