To find the numerical equivalent of the expression \( 9 + \frac{1}{9} - 27 - \frac{1}{27} + 27 - \frac{1}{9} \), we can simplify it step-by-step:
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Convert the fractions to a common denominator:
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The denominators are 9 and 27. The common denominator for both is 27.
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Convert \( 9 \): \[ 9 = \frac{9 \times 27}{27} = \frac{243}{27} \]
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Convert \( \frac{1}{9} \): \[ \frac{1}{9} = \frac{1 \times 3}{9 \times 3} = \frac{3}{27} \]
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\( -\frac{1}{27} \) remains as it is.
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Convert \( -\frac{1}{9} \): \[ -\frac{1}{9} = -\frac{3}{27} \]
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Rewrite the expression with common denominator: \[ \frac{243}{27} + \frac{3}{27} - 27 - \frac{1}{27} + 27 - \frac{3}{27} \]
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Combine the terms:
- Notice the \( -27 + 27 \) cancels out.
The expression simplifies to: \[ \frac{243}{27} + \frac{3}{27} - \frac{1}{27} - \frac{3}{27} \]
Which simplifies further: \[ \frac{243 + 3 - 1 - 3}{27} = \frac{242}{27} \]
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Calculate the Fraction: \( \frac{242}{27} \) is approximately \( 8.96 \) when calculated as a decimal.
Therefore, the numerical equivalent of \( 9 + \frac{1}{9} - 27 - \frac{1}{27} + 27 - \frac{1}{9} \) is:
\[ \frac{242}{27} \text{ (or approximately } 8.96 \text{)} \]