To identify which of the following numbers is irrational, let's analyze each option:
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3.6363636363636363... - This number is a repeating decimal, which can be expressed as a fraction (in this case, \( \frac{109}{30} \)). Therefore, it is rational.
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−7837 - This is an integer and can be expressed as a fraction (\( \frac{-7837}{1} \)), so it is rational.
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−7 Start Fraction 8 over 37 End Fraction - Assuming this means \(-7\frac{8}{37}\), this is a mixed number that can be converted to an improper fraction, making it rational.
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3√4 Start Fraction Start Root 3 End Root over 4 End Fraction - Assuming this is meant to read as \(\frac{\sqrt{3}}{4}\), where \(\sqrt{3}\) is indeed an irrational number, making the entire expression irrational.
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52.781654292 - This number is a non-repeating decimal. While it may appear to be a long decimal, if it does not repeat, it is still considered rational if it can be expressed as a fraction.
Given this analysis, the only irrational number in the list is \(\frac{\sqrt{3}}{4}\).