To express the repeating decimal \( 0.666\ldots0.666\ldots0.666\ldots \) as a fraction, we need to identify the repeating part.
Let's denote the decimal expansion as \( x \): \[ x = 0.666\ldots \]
The digit "6" repeats after the decimal indefinitely. To convert this repeating decimal into a fraction, we can set up the following equation:
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Multiply \( x \) by 10 (to shift the decimal point to the right): \[ 10x = 6.666\ldots \]
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Now, set up the equation using the original \( x \): \[ 10x = 6 + x \]
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Rearranging this gives: \[ 10x - x = 6 \] \[ 9x = 6 \]
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Solving for \( x \) yields: \[ x = \frac{6}{9} \]
This fraction can be simplified. However, you mentioned that simplification is not needed, so we can simply state the infinite decimal expansion as a fraction without simplification:
\[ \frac{6}{9} \]
Thus, the repeating decimal \( 0.666\ldots0.666\ldots0.666\ldots \) can be expressed as the fraction \( \frac{6}{9} \).
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