Let's analyze the given expression step by step. The expression is:
(32)×(32)‌×(32)‌×...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=25Thus, the original expression becomes:
(25)×(25)‌×(25)‌×...We can use the properties of exponents to combine the terms:
=25×2‌×2‌×...By adding the exponents, we obtain the combined exponent for the entire product:
=25+‌+‌+... Now, notice that this is an infinite series where each term is of the form
‌. The exponents form an infinite geometric series:
5+‌+‌+‌+...The first term
a of this geometric series is 5 , and the common ratio
r is
‌. The sum
S of an infinite geometric series is given by:
S=‌Substituting the values
a=5 and
r=‌, we get:
S=‌=‌=6Thus, the combined exponent for the product is 6 . Therefore, the final expression is:
26=64Hence, the given expression evaluates to 64 . The correct answer is:
Option D: 64.