Certainly! Let's examine each option step by step to determine which number is irrational.
Option A: [tex]\(\sqrt{100}\)[/tex]
The square root of 100 is:
[tex]\[
\sqrt{100} = 10
\][/tex]
10 is a whole number, which means it is rational. Therefore, [tex]\(\sqrt{100}\)[/tex] is not an irrational number.
Option B: [tex]\(\frac{1}{8}\)[/tex]
The fraction [tex]\(\frac{1}{8}\)[/tex] can be expressed as a decimal:
[tex]\[
\frac{1}{8} = 0.125
\][/tex]
Since 0.125 is a terminating decimal, it is rational. Therefore, [tex]\(\frac{1}{8}\)[/tex] is not an irrational number.
Option C: -2.2675
The number -2.2675 is a finite decimal, which can be expressed as a fraction:
[tex]\[
-2.2675 = -\frac{22675}{10000}
\][/tex]
Since it can be expressed as a fraction, it is rational. Therefore, -2.2675 is not an irrational number.
Option D: [tex]\(\sqrt[3]{16}\)[/tex]
The cube root of 16 is not a whole number. Let's consider its properties:
[tex]\[
\sqrt[3]{16} \approx 2.5198421...
\][/tex]
This value does not terminate or repeat, meaning it cannot be precisely expressed as a fraction. Therefore, [tex]\(\sqrt[3]{16}\)[/tex] is an irrational number.
After analyzing each option, we conclude that:
The correct answer is:
D. [tex]\(\sqrt[3]{16}\)[/tex]
