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Which of the following is a rational number?\newlineChoices:\newline(A) 27\frac{2}{7}\newline(B) Ο€\pi\newline(C) 3\sqrt{3}\newline(D) 8\sqrt{8}

Full solution

Q. Which of the following is a rational number?\newlineChoices:\newline(A) 27\frac{2}{7}\newline(B) Ο€\pi\newline(C) 3\sqrt{3}\newline(D) 8\sqrt{8}
  1. Definition of Rational Number: A rational number is a number that can be expressed as the quotient or fraction pq\frac{p}{q} of two integers, where pp and qq are integers and qq is not zero. We will examine each choice to determine if it can be expressed in this form.
  2. Choice (A): 27\frac{2}{7}: Choice (A) is 27\frac{2}{7}. This is already in the form of a fraction where both the numerator (22) and the denominator (77) are integers, and the denominator is not zero. Therefore, 27\frac{2}{7} is a rational number.
  3. Choice (B): Ο€\pi: Choice (B) is Ο€\pi (pi). Pi is a transcendental number, which means it cannot be expressed as a fraction of two integers. Therefore, Ο€\pi is not a rational number.
  4. Choice (C): 3\sqrt{3}: Choice (C) is the square root of 33. The square root of 33 is an irrational number because it cannot be expressed as a fraction of two integers. Its decimal expansion is non-terminating and non-repeating. Therefore, 3\sqrt{3} is not a rational number.
  5. Choice (D): 8\sqrt{8}: Choice (D) is the square root of 88. The square root of 88 simplifies to 22 times the square root of 22, which is an irrational number. Therefore, 8\sqrt{8} is not a rational number.

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