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The number vv is irrational. Which statement about v8v - \sqrt{8} is true?\newlineChoices:\newline(A)v8v - \sqrt{8} is rational.\newline(B)v8v - \sqrt{8} is irrational.\newline(C)v8v - \sqrt{8} can be rational or irrational, depending on the value of vv.

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Q. The number vv is irrational. Which statement about v8v - \sqrt{8} is true?\newlineChoices:\newline(A)v8v - \sqrt{8} is rational.\newline(B)v8v - \sqrt{8} is irrational.\newline(C)v8v - \sqrt{8} can be rational or irrational, depending on the value of vv.
  1. Identify Type of Number: Identify whether 8\sqrt{8} is a rational or irrational number. 88 is a non-perfect square, which means that its square root cannot be expressed as a simple fraction of two integers. Therefore, 8\sqrt{8} is an irrational number.
  2. Properties of Irrational Numbers: Consider the properties of irrational numbers. The sum or difference of an irrational number and a rational number is always irrational. However, the sum or difference of two irrational numbers can be either rational or irrational, depending on the specific numbers involved.
  3. Analyze v8v - \sqrt{8}: Analyze the possible outcomes for v8v - \sqrt{8}. If vv is an irrational number that is not related to 8\sqrt{8}, then v8v - \sqrt{8} will be irrational because the difference of two unrelated irrational numbers is irrational. If v=8v = \sqrt{8}, then v8v - \sqrt{8} = 88\sqrt{8} - \sqrt{8} = 00, which is rational. Therefore, v8v - \sqrt{8} can be rational or irrational, depending on the value of vv.

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