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.