To determine if the fraction [tex]\(\frac{17}{40}\)[/tex] represents a terminating decimal, we need to follow these steps:
1. Simplify the Fraction:
Firstly, we check if the fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 17 and 40 is 1. Therefore, the fraction [tex]\(\frac{17}{40}\)[/tex] is already in its simplest form.
2. Prime Factorization of the Denominator:
For the fraction to represent a terminating decimal, the simplified denominator must have prime factors of only 2 and/or 5. We perform the prime factorization of 40:
[tex]\[
40 = 2^3 \times 5
\][/tex]
Hence, the denominator 40 can be expressed as a product of the prime numbers 2 and 5.
3. Checking for Terminating Decimal:
Since the denominator has only the prime factors 2 and 5, it meets the criteria for the fraction to be a terminating decimal.
Given these conditions are satisfied, we conclude that [tex]\(\frac{17}{40}\)[/tex] indeed represents a terminating decimal.
Thus, the fraction [tex]\(\frac{17}{40}\)[/tex] does represent a terminating decimal.
