Answer :
To determine which number is an irrational number, let's analyze each option one by one:
A. [tex]\( \sqrt{100} \)[/tex]
First, we find the square root of 100:
[tex]\[ \sqrt{100} = 10 \][/tex]
Since 10 is a whole number, it is a rational number.
B. [tex]\( \frac{1}{8} \)[/tex]
Next, we examine the fraction [tex]\(\frac{1}{8}\)[/tex]:
[tex]\[ \frac{1}{8} = 0.125 \][/tex]
Since 0.125 is a terminating decimal, it is a rational number.
C. [tex]\( -2.2675 \)[/tex]
Consider the number -2.2675:
[tex]\[ -2.2675 \][/tex]
Since -2.2675 is a terminating decimal, it is also a rational number.
D. [tex]\( \sqrt[3]{16} \)[/tex]
Now, we look at the cube root of 16. The cube root of 16 cannot be simplified to an exact fraction or a terminating or repeating decimal. Thus, it does not fit the definition of a rational number.
Since an irrational number cannot be expressed as a fraction and cannot be a terminating or repeating decimal, the correct answer is:
[tex]\[ \sqrt[3]{16} \][/tex]
Hence, the irrational number is choice D.
A. [tex]\( \sqrt{100} \)[/tex]
First, we find the square root of 100:
[tex]\[ \sqrt{100} = 10 \][/tex]
Since 10 is a whole number, it is a rational number.
B. [tex]\( \frac{1}{8} \)[/tex]
Next, we examine the fraction [tex]\(\frac{1}{8}\)[/tex]:
[tex]\[ \frac{1}{8} = 0.125 \][/tex]
Since 0.125 is a terminating decimal, it is a rational number.
C. [tex]\( -2.2675 \)[/tex]
Consider the number -2.2675:
[tex]\[ -2.2675 \][/tex]
Since -2.2675 is a terminating decimal, it is also a rational number.
D. [tex]\( \sqrt[3]{16} \)[/tex]
Now, we look at the cube root of 16. The cube root of 16 cannot be simplified to an exact fraction or a terminating or repeating decimal. Thus, it does not fit the definition of a rational number.
Since an irrational number cannot be expressed as a fraction and cannot be a terminating or repeating decimal, the correct answer is:
[tex]\[ \sqrt[3]{16} \][/tex]
Hence, the irrational number is choice D.
