Let's find the fractional form of the repeating decimal [tex]\(0 . \overline{8}\)[/tex].
1. First, let's denote the repeating decimal by [tex]\(x\)[/tex]:
[tex]\[
x = 0.8888 \ldots
\][/tex]
2. Now, multiply both sides of this equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 8.8888 \ldots
\][/tex]
3. We now have two equations:
[tex]\[
\begin{aligned}
x &= 0.8888 \ldots \quad \text{(Equation 1)} \\
10x &= 8.8888 \ldots \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
4. Next, subtract Equation 1 from Equation 2 to eliminate the repeating part:
[tex]\[
\begin{aligned}
10x - x &= 8.8888 \ldots - 0.8888 \ldots \\
9x &= 8
\end{aligned}
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides of the equation by 9:
[tex]\[
x = \frac{8}{9}
\][/tex]
Therefore, the fractional form of [tex]\(0 . \overline{8}\)[/tex] is [tex]\(\frac{8}{9}\)[/tex].
So, the correct answer is:
C. [tex]\(\frac{8}{9}\)[/tex]
