Let's analyze each of the given numbers to determine which one is irrational.
1. [tex]\(4.5 \overline{71}\)[/tex]: This notation represents the repeating decimal [tex]\(4.5717171717\ldots\)[/tex]. Any repeating decimal can be expressed as a fraction of two integers, and hence it is a rational number.
2. [tex]\(-7.829\)[/tex]: This number is a terminating decimal. Any terminating decimal can also be expressed as a fraction of two integers. Hence, [tex]\(-7.829\)[/tex] is a rational number.
3. [tex]\(\sqrt{25}\)[/tex]: The square root of 25 is 5, because [tex]\(5 \times 5 = 25\)[/tex]. Since 5 is an integer, [tex]\(\sqrt{25}\)[/tex] is a rational number.
4. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]: The square root of 3 ([tex]\(\sqrt{3}\)[/tex]) is not an integer and cannot be precisely expressed as a fraction of two integers. Thus, [tex]\(\sqrt{3}\)[/tex] is an irrational number, and any multiple (or division in this case) of an irrational number by a rational number (like 2) also remains irrational. Therefore, [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is an irrational number.
Based on this analysis, the irrational number among the given options is:
[tex]\(\frac{\sqrt{3}}{2}\)[/tex]
