To determine which of the given numbers is irrational, let's analyze each option step by step:
A. [tex]\(\sqrt{25}\)[/tex]
- The square root of 25 is 5 because [tex]\(5 \times 5 = 25\)[/tex].
- Since 5 is a whole number, it is rational.
- Therefore, [tex]\(\sqrt{25}\)[/tex] is rational.
B. 0.45
- The decimal 0.45 is a terminating decimal, meaning it ends after a finite number of decimal places.
- Any terminating decimal can be expressed as a fraction (in this case, [tex]\(\frac{45}{100} = \frac{9}{20}\)[/tex]), which means it is rational.
- Therefore, 0.45 is rational.
C. [tex]\(\sqrt{6}\)[/tex]
- The square root of 6 cannot be simplified to a whole number or a simple fraction because 6 is not a perfect square.
- Numbers that cannot be expressed as a fraction of two integers and have non-repeating, non-terminating decimal expansions are irrational.
- Therefore, [tex]\(\sqrt{6}\)[/tex] is irrational.
D. [tex]\(0.636363 \ldots\)[/tex]
- The decimal [tex]\(0.636363 \ldots\)[/tex] is a repeating decimal because the sequence "63" repeats infinitely.
- Any repeating decimal can be expressed as a fraction (in this case, it can be written as [tex]\(\frac{63}{99}\)[/tex] and simplified further if needed), which means it is rational.
- Therefore, [tex]\(0.636363 \ldots\)[/tex] is rational.
Given the analysis:
- Option A: [tex]\(\sqrt{25}\)[/tex] is rational.
- Option B: 0.45 is rational.
- Option C: [tex]\(\sqrt{6}\)[/tex] is irrational.
- Option D: [tex]\(0.636363 \ldots\)[/tex] is rational.
Thus, the irrational number among the options is:
C. [tex]\(\sqrt{6}\)[/tex]
