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Select all the numbers that are irrational.\newlineMulti-select Choices:\newline(A) 34\sqrt{\frac{3}{4}}\newline(B) 343\sqrt{4}\newline(C) −34-\frac{3}{4}\newline(D) 0.34‾0.\overline{34}\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\sqrt{\frac{3}{4}}\newline(B) 343\sqrt{4}\newline(C) −34-\frac{3}{4}\newline(D) 0.34‾0.\overline{34}\newline(E) 34\sqrt{34}
  1. Analyze 34\sqrt{\frac{3}{4}}: Step 11: Analyze (A) 34\sqrt{\frac{3}{4}}.\newline34\sqrt{\frac{3}{4}} can be written as (3)/4(\sqrt{3})/4, which is an irrational number divided by a rational number. The result is irrational.
  2. Analyze 343\sqrt{4}: Step 22: Analyze (B) 343\sqrt{4}. 343\sqrt{4} simplifies to 3×2=63\times 2 = 6, which is a rational number.
  3. Analyze −34-\frac{3}{4}: Step 33: Analyze (C) −34-\frac{3}{4}.\newline−34-\frac{3}{4} is a rational number because it can be expressed as a fraction.
  4. Analyze 0.34‾0.\overline{34}: Step 44: Analyze (D) 0.34‾0.\overline{34}. 0.34‾0.\overline{34} represents a repeating decimal, which is rational.
  5. Analyze 34\sqrt{34}: Step 55: Analyze (E)34(E) \sqrt{34}.\newline34\sqrt{34} cannot be simplified to a rational number; it's an irrational number.

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