To determine which choice is equal to the repeating decimal [tex]\(0.\overline{3}\)[/tex], we need to analyze the options provided and see which one approximates the given repeating decimal [tex]\(0.\overline{3}\)[/tex].
First, let's understand what [tex]\(0.\overline{3}\)[/tex] means. The notation [tex]\(0.\overline{3}\)[/tex] represents the decimal 0.333333... where the digit 3 repeats infinitely.
Now, let's look at each option in detail:
Choice A: [tex]\(0.33333333 \ldots\)[/tex]
This option represents a decimal where the digit '3' repeats infinitely. It is another way to represent [tex]\(0.\overline{3}\)[/tex]. Therefore, this choice is exactly equal to the repeating decimal [tex]\(0.\overline{3}\)[/tex].
Choice B: 0.333
This is a finite decimal representation which stops after three decimal places. It approximates [tex]\(0.\overline{3}\)[/tex] but is not exactly the same because [tex]\(0.\overline{3}\)[/tex] extends infinitely beyond the decimal point, while 0.333 does not.
Choice C: [tex]\(\frac{3}{10}\)[/tex]
This fraction represents the decimal 0.3, which is not the same as [tex]\(0.\overline{3}\)[/tex]. The decimal [tex]\(0.3\)[/tex] is a terminating decimal and does not continue indefinitely like [tex]\(0.\overline{3}\)[/tex].
After comparing all the given options, we see that:
- Choice A: [tex]\(0.33333333 \ldots\)[/tex] is the exact representation of [tex]\(0.\overline{3}\)[/tex].
- Choice B (0.333) and Choice C ([tex]\(\frac{3}{10}\)[/tex]) are not exact matches and therefore incorrect.
Hence, the choice that is equal to the repeating decimal [tex]\(0.\overline{3}\)[/tex] is:
A. [tex]\(0.33333333 \ldots\)[/tex]
