To solve the problem of reducing the fraction [tex]\(\frac{27}{54}\)[/tex] into its simplest form and ensuring that the fraction is represented in its prime factored form, we can follow these steps:
1. Prime Factorization of the Numerator (27):
The first step is to determine the prime factorization of 27.
27 can be broken down into its prime factors as follows:
[tex]\[
27 = 3 \times 3 \times 3 = 3^3
\][/tex]
So, the prime factors of 27 are [tex]\([3]\)[/tex].
2. Prime Factorization of the Denominator (54):
Next, we need the prime factorization of 54.
54 can be expressed as:
[tex]\[
54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^2
\][/tex]
So, the prime factors of 54 are [tex]\([2, 3]\)[/tex].
3. Reduce the Fraction to its Simplest Form:
Now, we need to reduce the fraction [tex]\(\frac{27}{54}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor (GCD).
The GCD of 27 and 54 can be identified by looking at their common prime factors. Since both 27 and 54 have the prime factor 3, and 3 is the highest common factor, it is their GCD.
Given this, we divide the numerator and the denominator by 27:
[tex]\[
\frac{27}{54} = \frac{27 \div 27}{54 \div 27} = \frac{1}{2}
\][/tex]
4. Presenting the Fraction in Reduced Form:
After performing the division by the GCD, the reduced form of [tex]\(\frac{27}{54}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
Therefore, the correct simplified form of [tex]\(\frac{27}{54}\)[/tex] while showing its prime factors and reducing it to the simplest form is:
[tex]\[
\frac{1}{2}
\][/tex]
So, based on the options given, the correct answer is:
A. [tex]\(\frac{1}{2}\)[/tex]
