Question
Which of the following is a rational number? \( \sqrt{72} \) \( \sqrt{27} \) \( \sqrt{31} \) 30.666... Submits
Ask by Maxwell Davey. in the United States
Feb 28,2025
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Tutor-Verified Answer
Answer
30.666... is a rational number.
Solution
To determine which number is rational, we need to review each option:
• √72 = √(36×2) = 6√2, which is irrational because √2 is irrational.
• √27 = √(9×3) = 3√3, which is irrational because √3 is irrational.
• √31 is irrational since 31 is not a perfect square.
• 30.666... represents a number with a repeating (or terminating) decimal expansion. In fact, 30.666... is equivalent to 30 + 2/3, which can be expressed as the fraction 92/3, a ratio of two integers.
Thus, the rational number among the choices is 30.666....
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Simplify this solution Beyond the Answer
Did you know that rational numbers can be expressed as a fraction of two integers? This means any number that can be put in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \), fits the definition! For example, \( 30.666...\) can be represented as \( \frac{92}{3} \), confirming its rationality. A common mistake when identifying rational numbers is assuming that only whole numbers qualify. However, this is a myth! Decimals like \( 30.666...\) are indeed rational. On the contrary, the square roots in the options above, like \( \sqrt{72} \) and \( \sqrt{31} \), yield irrational results when calculated, so they won't fit the rational mold!
