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Is (36)/(25)*sqrt11 rational or irrational? Choose 1 answer: (A) Rational (B) Irrational C) It can be either rational or irrational

Is 3625⋅11\frac{36}{25} \cdot \sqrt{11} rational or irrational?\newlineChoose 11 answer:\newline(A) Rational\newline(B) Irrational\newline(C) It can be either rational or irrational

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Full solution

Q. Is 3625⋅11\frac{36}{25} \cdot \sqrt{11} rational or irrational?\newlineChoose 11 answer:\newline(A) Rational\newline(B) Irrational\newline(C) It can be either rational or irrational
  1. Understand rational vs irrational: Understand the definition of rational and irrational numbers.\newlineA 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. An irrational number cannot be expressed as a simple fraction; it's a number that cannot be written as a ratio of two integers.
  2. Analyze given expression: Analyze the given expression.\newlineThe expression is (3625)11(\frac{36}{25})\sqrt{11}. We know that 3636 and 2525 are integers, so (3625)(\frac{36}{25}) is a rational number. However, we need to determine the nature of 11\sqrt{11}.
  3. Determine 11\sqrt{11} nature: Determine the nature of 11\sqrt{11}. The square root of 1111, 11\sqrt{11}, is an irrational number because 1111 is not a perfect square, and its square root cannot be expressed as a fraction of two integers.
  4. Multiply rational by irrational: Multiply a rational number by an irrational number.\newlineWhen we multiply a rational number by an irrational number, the result is always an irrational number. This is because the irrational number cannot be expressed as a fraction, and multiplying by a fraction (rational number) does not change this fact.
  5. Conclude expression type: Conclude whether the expression is rational or irrational.\newlineSince (3625)(\frac{36}{25}) is rational and 11\sqrt{11} is irrational, their product (3625)⋅11(\frac{36}{25})\cdot\sqrt{11} is irrational.

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