To determine which of the given responses is irrational, let's analyze each option:
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\( \sqrt[3]{4} \): The cube root of 4 is an irrational number since it cannot be expressed as a fraction of two integers.
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\( 3.6363636363636363... \): This is a repeating decimal (that can be expressed as \( \frac{11}{3} \)), which is a rational number.
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\( -7837 \frac{8}{37} \): This is a rational number since both the integer part (-7837) and the fractional part (\( \frac{8}{37} \)) can be expressed as a fraction.
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\( 52.781654292 \): This is a finite decimal and can be represented as a fraction, so it is also a rational number.
Based on this analysis, the irrational number among the options is:
- \( \sqrt[3]{4} \).