Certainly! Let's go through each option one by one to determine whether it's rational or irrational.
1. Option A: [tex]\(\sqrt{100}\)[/tex]
[tex]\[
\sqrt{100} = 10
\][/tex]
The square root of 100 is 10, which is an integer. An integer is a rational number because it can be expressed as the fraction [tex]\(\frac{10}{1}\)[/tex].
2. Option B: [tex]\(\frac{1}{8}\)[/tex]
[tex]\[
\frac{1}{8} = 0.125
\][/tex]
The fraction [tex]\(\frac{1}{8}\)[/tex] is a rational number because it is a ratio of two integers (1 and 8) and can be expressed as a terminating decimal, 0.125.
3. Option C: [tex]\(-2.2675\)[/tex]
[tex]\[
-2.2675
\][/tex]
The number [tex]\(-2.2675\)[/tex] is a finite decimal. Finite decimals are rational because they can be written as a fraction. For instance, [tex]\(-2.2675\)[/tex] can be expressed as [tex]\(-\frac{22675}{10000}\)[/tex].
4. Option D: [tex]\(\sqrt[3]{16}\)[/tex]
[tex]\(\sqrt[3]{16}\)[/tex] is the cube root of 16. To determine if it's rational or irrational, consider whether it can be written as a fraction or has a terminating or repeating decimal representation.
The cube root of 16 is not a nice integer value and cannot be expressed as a simple fraction. Its decimal representation is non-terminating and non-repeating. Therefore, it is an irrational number.
After examining all the options, we see that:
- Option A is rational.
- Option B is rational.
- Option C is rational.
- Option D is irrational.
Hence, the correct answer is:
[tex]\[
D. \sqrt[3]{16}
\][/tex]
