(32) (32)^ 1 6 (32)^ 1 36 ----- is equal to

Step-by-step solution

Let the given expression be S = (32) (32)^ 1 6 (32)^ 1 36 . The expression can be written as S = (32)^ 1 + 1 6 + 1 36 + . The exponent is an infinite geometric series with first term a = 1 and common ratio r = 1 6 . The sum of an infinite geometric series is given by a 1 - r , provided |r| Substituting the values, the sum of the exponent is 1 1 - 1 6 = 1 5 6 = 6 5 . Therefore, S = (32)^ 6 5 . Since 32 = 2^5 , we have S = (2^5)^ 6 5 = 2^ 5 6 5 = 2^6 . 2^6 = 64 . Answer: 64