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Select all the numbers that are irrational.\newlineMulti-select Choices:\newline(A) −34-\frac{3}{4}\newline(B) 34\sqrt{\frac{3}{4}}\newline(C) 0.34‾0.\overline{34}\newline(D) 343\sqrt{4}\newline(E) 34\sqrt{34}

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Select all the numbers that are irrational.\newlineMulti-select Choices:\newline(A) −34-\frac{3}{4}\newline(B) 34\sqrt{\frac{3}{4}}\newline(C) 0.34‾0.\overline{34}\newline(D) 343\sqrt{4}\newline(E) 34\sqrt{34}
  1. Analyze Rational Number: Step 11: Analyze choice (A) −34-\frac{3}{4}. −34-\frac{3}{4} is a rational number because it can be expressed as a fraction of two integers.
  2. Analyze Irrational Number: Step 22: Analyze choice (B)34(B) \sqrt{\frac{3}{4}}.34\sqrt{\frac{3}{4}} simplifies to 3/2\sqrt{3}/2, which is irrational because the square root of 33 is an irrational number.
  3. Analyze Repeating Decimal: Step 33: Analyze choice (C) 0.34‾0.\overline{34}. 0.34‾0.\overline{34} represents a repeating decimal, which is rational because it can be expressed as a fraction.
  4. Analyze Rational Number: Step 44: Analyze choice (D) 343\sqrt{4}. 343\sqrt{4} simplifies to 3×2=63\times 2 = 6, which is a rational number.
  5. Analyze Irrational Number: Step 55: Analyze choice (E) 34\sqrt{34}.34\sqrt{34} is an irrational number because 3434 is not a perfect square, and its square root cannot be expressed as a fraction.

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