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Select all the numbers that are rational.\newlineMulti-select Choices:\newline(A)18\sqrt{18}\newline(B)−0.81-0.81\newline(C)0.18‾0.\overline{18}\newline(D)−81-\sqrt{81}\newline(E)81\frac{\sqrt{8}}{\sqrt{1}}

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Checkpoint: Rational and irrational numbersright chevron icon

Full solution

Q. Select all the numbers that are rational.\newlineMulti-select Choices:\newline(A)18\sqrt{18}\newline(B)−0.81-0.81\newline(C)0.18‾0.\overline{18}\newline(D)−81-\sqrt{81}\newline(E)81\frac{\sqrt{8}}{\sqrt{1}}
  1. Analyze 18\sqrt{18}: Step 11: Analyze (A) 18\sqrt{18}.18=9×2=3×2\sqrt{18} = \sqrt{9\times2} = 3\times\sqrt{2}.2\sqrt{2} is known to be irrational; therefore, 3×23\times\sqrt{2} is also irrational.
  2. Analyze −0.81-0.81: Step 22: Analyze (B) −0.81-0.81.\newline−0.81-0.81 is a decimal that terminates.\newlineTerminating decimals are rational numbers.
  3. Analyze 0.18‾0.\overline{18}: Step 33: Analyze (C) 0.18‾0.\overline{18}. 0.18‾0.\overline{18} represents the repeating decimal 0.181818…0.181818\ldots Repeating decimals are rational numbers.
  4. Analyze −81-\sqrt{81}: Step 44: Analyze (D) −81-\sqrt{81}.−81=−9.-\sqrt{81} = -9.−9-9 is a whole number, and all whole numbers are rational.
  5. Analyze 81\sqrt{\frac{8}{1}}: Step 55: Analyze (E) 81\sqrt{\frac{8}{1}}.81=81=8\sqrt{\frac{8}{1}} = \frac{\sqrt{8}}{1} = \sqrt{8}.8=4⋅2=2⋅2\sqrt{8} = \sqrt{4\cdot2} = 2\cdot\sqrt{2}.Since 2\sqrt{2} is irrational, 2⋅22\cdot\sqrt{2} is also irrational.

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