Let's match each of the given numbers with its correct description (rational or irrational) and explain why.
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\( \sqrt{101} \)
- Description: Irrational
- Why: \( \sqrt{101} \) cannot be expressed as a fraction of two integers because 101 is not a perfect square, and its decimal representation is non-repeating and non-terminating.
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0.424242424...
- Description: Rational
- Why: This number has a repeating decimal (the "42" repeats), which can be expressed as a fraction (specifically \( \frac{42}{99} \)).
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0.133134135...
- Description: Irrational
- Why: The decimal appears to be non-repeating and non-terminating. As such, it cannot be expressed as a fraction of two integers.
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\( \sqrt{16} \)
- Description: Rational
- Why: \( \sqrt{16} = 4 \), which is an integer and can be expressed as a fraction (\( \frac{4}{1} \)).
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0.789
- Description: Rational
- Why: This number has a finite decimal representation, which can be expressed as a fraction (\( \frac{789}{1000} \)).
To summarize:
- \( \sqrt{101} \): Irrational
- 0.424242424...: Rational
- 0.133134135...: Irrational
- \( \sqrt{16} \): Rational
- 0.789: Rational