To convert the repeating decimal \(0.09\ldots\) into a rational number in simplest form, we'll follow these steps:
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Let \(x = 0.09\ldots\). \[ x = 0.090909\ldots \]
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Since the repeating part "09" has a length of 2, we can multiply by \(100\) to shift the decimal point two places to the right: \[ 100x = 9.090909\ldots \]
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Now we have two equations:
- \(x = 0.090909\ldots\)
- \(100x = 9.090909\ldots\)
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Now we'll subtract the first equation from the second: \[ 100x - x = 9.090909\ldots - 0.090909\ldots \] \[ 99x = 9 \]
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Now, divide both sides by \(99\): \[ x = \frac{9}{99} \]
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Simplify the fraction \(\frac{9}{99}\): \[ x = \frac{9 \div 9}{99 \div 9} = \frac{1}{11} \]
Thus, the repeating decimal \(0.09\ldots\) can be expressed as the rational number \(\frac{1}{11}\) in simplest form.
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