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The number v v is irrational. Which statement about v47 v - \sqrt{47} is true?\newlineChoices:\newline(A)v47 v - \sqrt{47} is rational.\newline(B)v47 v - \sqrt{47} is irrational.\newline(C)v47 v - \sqrt{47} can be rational or irrational, depending on the value of v v .

Full solution

Q. The number v v is irrational. Which statement about v47 v - \sqrt{47} is true?\newlineChoices:\newline(A)v47 v - \sqrt{47} is rational.\newline(B)v47 v - \sqrt{47} is irrational.\newline(C)v47 v - \sqrt{47} can be rational or irrational, depending on the value of v v .
  1. Identify Type of Number: Identify whether 47\sqrt{47} is a rational or irrational number.4747 is a non-perfect square.47\sqrt{47} 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 irrational. The sum or difference of two irrational numbers can be rational if they are additive inverses of each other, or irrational otherwise.
  3. Analyze Given Expressions: Analyze the given expressions with respect to the properties of irrational numbers. \newlinevv is an irrational number. \newline47\sqrt{47} is an irrational number. \newlineIf v=47v = \sqrt{47}, then v47=0v - \sqrt{47} = 0, which is rational. \newlineIf v47v \neq \sqrt{47}, then v47v - \sqrt{47} is irrational. \newlineTherefore, v47v - \sqrt{47} can be rational or irrational, depending on the value of vv.

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