Okay, let's determine which of the given numbers is a rational number. A rational number is defined as a number that can be expressed as the quotient or fraction [tex]\( \frac{p}{q} \)[/tex] of two integers, where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
Let's analyze each of the given options one by one:
A. [tex]\( \sqrt{15} \)[/tex]
- The square root of 15 is an irrational number because 15 is not a perfect square, and its square root cannot be expressed as a precise fraction of two integers. It has a non-repeating, non-terminating decimal expansion.
B. 2.6457513110...
- This number appears to be an approximation of a decimal that goes on indefinitely without repeating a pattern. Such a number cannot be expressed as a fraction, and hence it is irrational.
C. 17.156
- At first glance, it may seem rational because it is a decimal number, but not all decimal numbers are rational. Decimal numbers are considered rational if they terminate (like 4.5) or repeat (like 0.333...). Since 17.156 does not fall into these categories, it requires further analysis. However, from the context, it indicates that it is not expressible as a simple fraction, hence it is not rational.
D. 85
- The number 85 is a whole number and can be written as [tex]\( \frac{85}{1} \)[/tex], where 85 and 1 are both integers and 1 is not zero.
Based on the above analysis, the only rational number among the given options is:
D. 85
Thus, the correct answer is option D.
