simplify

Question

To convert the repeating decimal \(0.09\overline{9}\) (where the \(9\) repeats indefinitely) into a rational number, we can set \(x\) equal to the decimal:

\[
x = 0.099999\ldots
\]

Next, we multiply both sides by \(10\) to shift the decimal point one place to the right:

\[
10x = 0.99999\ldots
\]

Now we have two equations:

1. \(x = 0.099999\ldots\)
2. \(10x = 0.99999\ldots\)

Next, notice that the repeating part in equation \(10x\) is the same as that in \(x\). We can rewrite \(0.99999\ldots\) as \(1\):

\[
10x = 1
\]

Now we can solve for \(x\):

\[
x = \frac{1}{10}
\]

Thus, we have:

\[
0.09\overline{9} = \frac{1}{10}
\]

The rational number in simplest form is:

\[
\boxed{\frac{1}{10}}
\]

Answer 1

Let's go through the simplification of the repeating decimal \(0.09\overline{9}\) step by step to convert it into a rational number.

We start by letting:

\[ x = 0.099999\ldots \]

Next, we multiply both sides by \(10\) to move the decimal point one place to the right:

\[ 10x = 0.99999\ldots \]

Now we have the two equations:

1. \(x = 0.099999\ldots\)
2. \(10x = 0.99999\ldots\)

Note that the expression \(0.99999\ldots\) is equivalent to \(1\). Thus, we can rewrite our second equation as:

\[ 10x = 1 \]

Now, we can solve for \(x\):

\[ x = \frac{1}{10} \]

This shows that:

\[ 0.09\overline{9} = \frac{1}{10} \]

In simplest form, the rational number representation is:

\[ \boxed{\frac{1}{10}} \]