To determine which of the given numbers is irrational, we need to analyze each option:
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3.6363636363636363... - This is a repeating decimal (specifically, it repeats the "63" indefinitely). Repeating decimals can be expressed as fractions, which means they are rational numbers.
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52.781654292 - This is a non-repeating decimal. However, non-repeating decimals can either be rational or irrational. In this case, it is a finite decimal, which can also be expressed as a fraction. Therefore, it is rational.
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3√4 - This notation seems to intend to represent the cube root of 4. The cube root of 4 is approximately 1.5874, and since it cannot be expressed as a fraction of two integers, it is considered an irrational number.
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−7837 - This is a whole number, and all integers are rational numbers.
Now, looking at the options, 3√4 (or the cube root of 4) is the only irrational number among the choices provided.
So, the answer is:
3√4 (cube root of 4).