COMEDK 2024 Shift 1 Paper

Let's analyze the given expression step by step. The expression is:
(32)×(32)‌

1

6

×(32)‌

1

36

×...
First, we observe that the base of each term in the product is 32 , which can be written as 25. Let's rewrite the expression using this form:
32=25
Thus, the original expression becomes:

(25)×(25)‌

1

6

×(25)‌

1

35

×...
We can use the properties of exponents to combine the terms:
=25×2‌

5

6

×2‌

5

36

×...
By adding the exponents, we obtain the combined exponent for the entire product:
=25+‌

5

6

+‌

5

36

+...

Now, notice that this is an infinite series where each term is of the form ‌

5

6n

. The exponents form an infinite geometric series:
5+‌

5

6

+‌

5

36

+‌

5

216

+...
The first term a of this geometric series is 5 , and the common ratio r is ‌

1

6

. The sum S of an infinite geometric series is given by:

S=‌

a

1−r

Substituting the values a=5 and r=‌

1

6

, we get:
S=‌

5

1−‌ 1 6

=‌

5

5 6

=6
Thus, the combined exponent for the product is 6 . Therefore, the final expression is:
26=64
Hence, the given expression evaluates to 64 . The correct answer is:
Option D: 64.