To determine which of the given options is a rational number, we need to understand the definition of rational and irrational numbers.
A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, it can be written in the form [tex]\( \frac{a}{b} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex].
An irrational number, on the other hand, cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating.
Let's analyze each option in detail:
1. [tex]\( \sqrt{7} \)[/tex]:
- The number [tex]\( 7 \)[/tex] is not a perfect square.
- Therefore, [tex]\( \sqrt{7} \)[/tex] cannot be expressed as a fraction of two integers.
- Hence, [tex]\( \sqrt{7} \)[/tex] is an irrational number.
2. [tex]\( \frac{8}{\sqrt{8}} \)[/tex]:
- Simplifying [tex]\( \frac{8}{\sqrt{8}} \)[/tex]:
[tex]\[
\frac{8}{\sqrt{8}} = \frac{8}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{8\sqrt{8}}{8} = \sqrt{8}
\][/tex]
- Since [tex]\( \sqrt{8} \)[/tex] is not a perfect square, it cannot be expressed as a fraction of two integers.
- Therefore, [tex]\( \frac{8}{\sqrt{8}} \)[/tex] is also an irrational number.
3. 7.8:
- The decimal number 7.8 can be expressed as the fraction [tex]\( \frac{78}{10} \)[/tex], where both 78 and 10 are integers.
- Since this is a simple fraction of integers, 7.8 is a rational number.
4. [tex]\( \pi \)[/tex]:
- The number [tex]\( \pi \)[/tex] (pi) is known to be irrational.
- It cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating.
Based on the analysis:
- [tex]\( \sqrt{7} \)[/tex] is irrational.
- [tex]\( \frac{8}{\sqrt{8}} \)[/tex] is irrational.
- 7.8 is rational.
- [tex]\( \pi \)[/tex] is irrational.
Thus, the rational number among the options provided is 7.8.
