15.77 is a rational number. A rational number is any number that can be expressed as a fraction where the numerator and denominator are integers and the denominator is not zero. In this case, 15.77 can be written as 1577/100, which meets the criteria for a rational number.
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10.01 is a rational number
Rational
Such a product is always irrational - unless the rational number happens to be zero.
The sum of a rational and irrational number must be an irrational number.
rational * irrational = irrational.