To determine which of the given options is irrational, let's analyze each one:
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3√4 - This means the cube root of 4. The cube root of any rational number is also rational, so this is rational.
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Start Fraction Start Root 3 End Root over 4 End Fraction - This represents \(\frac{\sqrt{3}}{4}\). Since \(\sqrt{3}\) is an irrational number, \(\frac{\sqrt{3}}{4}\) is also irrational.
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3.6363636363636363... - This is a repeating decimal, which makes it a rational number.
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52.781654292 - This is a non-repeating, non-terminating decimal. However, if it has a finite number of decimal places, it is rational. Since we don't know if it continues indefinitely or not from the representation, but typically, this would be treated as rational.
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−7837 - This is a whole number and therefore rational.
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Start Fraction 8 over 37 End Fraction - This fraction represents \(\frac{8}{37}\), which is a rational number.
Given this breakdown, the option that is irrational is:
Start Fraction Start Root 3 End Root over 4 End Fraction (or \(\frac{\sqrt{3}}{4}\)).