To simplify the fraction [tex]\(\frac{127}{256}\)[/tex], we begin by checking if there are any common factors between the numerator (127) and the denominator (256) other than 1.
1. Prime Factorization:
- First, we note that 127 is a prime number. A prime number has no divisors other than 1 and itself.
- Next, let's consider the prime factorization of 256. [tex]\(256 = 2^8\)[/tex], meaning 256 is a power of 2.
2. Greatest Common Divisor (GCD):
- Since 127 is a prime number and 256's factors are only powers of 2, 127 does not share any common factors with 256.
- The only common factor between 127 and 256 is 1.
3. Conclusion:
- Because the greatest common divisor (GCD) of 127 and 256 is 1, the fraction [tex]\(\frac{127}{256}\)[/tex] is already in its simplest form.
Therefore, the simplified form of the fraction [tex]\(\frac{127}{256}\)[/tex] is [tex]\(\frac{127}{256}\)[/tex].
