To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by what number? (0.264 is a repeating number, the numbers that repeat are 264.)

Answer 1

To convert the repeating decimal \( x = 0.264\overline{264} \) into a rational number, you'll want to eliminate the repeating part. The repeating part "264" has three digits.

To do this, you should multiply both sides of the equation by \( 1000 \) (which is \( 10^3 \)) because the repeating block has three digits. This will shift the decimal point three places to the right.

Here’s how the steps look:

1. Set \( x = 0.264\overline{264} \).
2. Multiply both sides by \( 1000 \): \[ 1000x = 264.\overline{264} \]

Now, the equation becomes:

• \( x = 0.264\overline{264} \)
• \( 1000x = 264.\overline{264} \)

Next, you can subtract the first equation from the second: \[ 1000x - x = 264.\overline{264} - 0.264\overline{264} \] This simplifies to: \[ 999x = 264 \]

Finally, divide both sides by 999: \[ x = \frac{264}{999} \]

To fully convert 0.264̅ into a rational number, you would multiply both sides by 1000.