Answer:
OPTION A
OPTION B
OPTION C
Step-by-step explanation:
Irrational numbers are the subset of real numbers. Their decimal representation neither form a pattern nor terminate.
OPTION A: [tex]$ \sqrt{\frac{1}{2}} $[/tex]
This is equal to [tex]$ \frac{1}{\sqrt{2}} $[/tex].
[tex]$ \sqrt{2} = 1.414... $[/tex] is non-terminating. So, it is an irrational number. Hence, the reciprocal of an irrational number would also be irrational. So, OPTION A is irrational.
OPTION B: [tex]$ \sqrt{\frac{1}{8}} $[/tex]
This is equal to [tex]$ \frac{1}{2\sqrt{2}} $[/tex]. Using the same logic as Option A, we regard OPTION B to be irrational as well.
OPTION C: [tex]$ \sqrt{\frac{1}{10}} $[/tex]
This is equal to [tex]$ \frac{1}{\sqrt{5}\sqrt{2}} $[/tex].
Both [tex]$ \sqrt{5} $[/tex] and [tex]$ \sqrt{2} $[/tex] are irrational. So, the product and the reciprocal of the product is irrational as well. So, OPTION C is an irrational number.
OPTION D: [tex]$ \sqrt{\frac{1}{16}} $[/tex]
16 is a perfect square and is a rational number. [tex]$ \frac{1}{\sqrt{16}} $[/tex] = [tex]$ \frac{1}{4} $[/tex]. This is equal to 0.25, a terminating decimal. So, OPTION D is a rational number.
OPTION E: [tex]$ \sqrt{\frac{1}{4}} $[/tex]
4 is a perfect square as well. [tex]$ \frac{1}{\sqrt{4}} = \frac{1}{2} = 0.5 $[/tex], a terminating decimal. So, OPTION E is a rational number.
