The polymerase chain reaction (PCR) is an in vitro technique used to amplify DNA. During each cycle of PCR, the amount of DNA doubles. The total amount of DNA after \( n \) cycles is given by the equation \( 2^n \), where \( n \) is the number of cycles. Therefore, the number of DNA copies after each PCR cycle increases exponentially as \( 2^n \).
Thus, the correct answer is \( 2^n \).