Let \( x = 0.896896896\ldots \).
To express this repeating decimal as a fraction, we can identify the repeating part. The repeating block is "896", which has a length of 3. We can manipulate \( x \) as follows:
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Multiply \( x \) by \( 1000 \) (since there are 3 digits in the repeating part): \[ 1000x = 896.896896896\ldots \]
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Now, we have two equations: \[ x = 0.896896896\ldots \quad (1) \] \[ 1000x = 896.896896896\ldots \quad (2) \]
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Subtract equation (1) from equation (2): \[ 1000x - x = 896.896896896\ldots - 0.896896896\ldots \] \[ 999x = 896 \]
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Solving for \( x \): \[ x = \frac{896}{999} \]
So, the repeating decimal \( 0.896896896\ldots \) can be expressed as the fraction \( \frac{896}{999} \).