To convert 0.264¯¯¯¯¯¯¯¯
0.
264
¯
to a rational number, you would set x
𝑥
equal to 0.264¯¯¯¯¯¯¯¯
0.
264
¯
and then multiply both sides of the equation by what number?

Answers

There are no human answers yet.
Answered by GPT-4o mini
To convert the repeating decimal \( x = 0.264\overline{264} \) to a rational number, you'll want to eliminate the repeating part. In this case, the digits "264" repeat.

### Step 1: Set up the equation
Let \( x = 0.264264264\ldots \)

### Step 2: Determine the multiplier
Since "264" has 3 digits, you will multiply both sides of the equation by \( 1000 \) (which is \( 10^3 \)) to shift the decimal point 3 places to the right:

\[
1000x = 264.264264264\ldots
\]

### Step 3: Set up a second equation
Now, you also have the original value of \( x \):

\[
x = 0.264264264\ldots
\]

### Step 4: Subtract the two equations
Now you can subtract the two equations:

\[
1000x - x = 264.264264264\ldots - 0.264264264\ldots
\]

This simplifies to:

\[
999x = 264
\]

### Step 5: Solve for \( x \)
Now, divide both sides by 999:

\[
x = \frac{264}{999}
\]

### Conclusion
So, to convert \( 0.264\overline{264} \) to a rational number, you multiply both sides of the equation by **1000**.