To determine which number is irrational from the provided options, let's analyze each one:
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3.6363636363636363...: This is a repeating decimal, which can be expressed as the fraction \( \frac{363636}{100000} \) or simplified, making it rational.
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3.6363636363636363... − 7837: Since \( 3.6363636363636363... \) is rational and subtracting a rational number (−7837) from it results in a rational number, this is also rational.
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−7: This is an integer, which is also a rational number.
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\( \frac{8}{37} \): This is a fraction of two integers, so it is rational.
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52.781654292: This is a terminating decimal, which can be expressed as a fraction, hence it is rational.
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\( 3 \sqrt{4} \): This simplifies to \( 3 \times 2 = 6 \), which is also rational.
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\( \frac{\sqrt{3}}{4} \): While both parts of the fraction are real numbers, since \( \sqrt{3} \) is irrational, the entire expression \( \frac{\sqrt{3}}{4} \) is irrational because it cannot be expressed as a fraction of two integers.
Based on the above analysis, the only irrational number from the options is \( \frac{\sqrt{3}}{4} \).