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

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Q. The number vv is irrational. Which statement about 32v\sqrt{32} - v is true?\newlineChoices:\newline(A) 32v\sqrt{32} - v is rational.\newline(B) 32v\sqrt{32} - v is irrational.\newline(C) 32v\sqrt{32} - v can be rational or irrational, depending on the value of vv.
  1. Simplify 32\sqrt{32}: We need to determine the nature of the expression 32v\sqrt{32} - v. First, let's simplify 32\sqrt{32}.\newline32\sqrt{32} can be simplified to 16×2\sqrt{16 \times 2}, which is 16×2\sqrt{16} \times \sqrt{2}.\newlineSince 16\sqrt{16} is 44 and 2\sqrt{2} is an irrational number, 32\sqrt{32} simplifies to 32v\sqrt{32} - v00.
  2. Express as 42v4 \sqrt{2} - v: Now we have the expression 42v4 \sqrt{2} - v. Since 424 \sqrt{2} is irrational (because 2\sqrt{2} is irrational), and vv is also irrational, we need to consider the subtraction of two irrational numbers.\newlineThe subtraction of two irrational numbers can be either rational or irrational. There is no definitive rule that guarantees the result will be one or the other without knowing the specific values of the irrational numbers.
  3. Consider rationality of result: However, since we do not have any specific information about the value of vv other than it is irrational, we cannot determine if 4×2v4 \times \sqrt{2} - v will be rational or irrational. The result could be rational if vv happens to be exactly 4×24 \times \sqrt{2}, but in all other cases, it will be irrational.\newlineTherefore, without additional information about vv, we cannot definitively say that 32v\sqrt{32} - v is rational or irrational.

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