Sure! Let's write the infinite decimal expansion [tex]\(0.\overline{7}\)[/tex] (which means 0.777...) as a fraction. Here are the steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 0.\overline{7}
\][/tex]
2. Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 7.\overline{7}
\][/tex]
3. Now, subtract the original equation from this new equation to eliminate the repeating part:
[tex]\[
10x - x = 7.\overline{7} - 0.\overline{7}
\][/tex]
4. Simplify the subtraction:
[tex]\[
9x = 7
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{7}{9}
\][/tex]
Therefore, the fraction representation of the repeating decimal [tex]\(0.\overline{7}\)[/tex] is [tex]\(\frac{7}{9}\)[/tex].
