I have to find the prime factorization of each number and write down the answers with exponents when repeated factors appear.

eighteen, four hundred eighty, six hundred forty

Next, I need to write this fraction in lowest terms.

one-hundred sixty-five over one-hundred eighty

This is not hard. Just start with 2, and divide it into the number as many times as you can. Then try 3,5,7,... for all primes up to sqrt(N). Why stop there? because if one factis greater than sqrt(N), the other factor will be less.

18
2*9
2*3*3 = 2*3^2

480
2*240
2*2*120
2*2*2*60
2*2*2*2*30
2^5 * 15
2^5 * 3 * 5

640
2^6 * 10
2^7 * 5

165/180

3*5*11/2*2*3*3*5
11/2*2*3
11/12

To find the prime factorization of a number, you need to decompose it into a product of prime numbers. Here's how you can find the prime factorization for each of the given numbers:

1) Eighteen (18):
Start by dividing 18 by the smallest prime number, which is 2. Since 18 is divisible by 2, we can write it as:
18 = 2 × 9
Now, focus on the quotient, 9, and continue the process. Since 9 is divisible by 3, we can further write it as:
18 = 2 × 3 × 3
Therefore, the prime factorization of 18 is 2 × 3 × 3, or 2² × 3.

2) Four hundred eighty (480):
Begin by dividing 480 by 2, as it is an even number divisible by 2:
480 = 2 × 240
Next, divide 240 by 2:
480 = 2 × 2 × 120
Then, divide 120 by 2:
480 = 2 × 2 × 2 × 60
Continue dividing 60 by 2:
480 = 2 × 2 × 2 × 2 × 30
Since 30 is divisible by 2, divide it again:
480 = 2 × 2 × 2 × 2 × 2 × 15
Finally, divide 15 by its smallest prime factor, which is 3:
480 = 2 × 2 × 2 × 2 × 2 × 3 × 5
Thus, the prime factorization of 480 is 2⁵ × 3 × 5.

3) Six hundred forty (640):
Start by dividing 640 by 2:
640 = 2 × 320
Divide 320 by 2:
640 = 2 × 2 × 160
Divide 160 by 2:
640 = 2 × 2 × 2 × 80
Divide 80 by 2:
640 = 2 × 2 × 2 × 2 × 40
Divide 40 by 2:
640 = 2 × 2 × 2 × 2 × 2 × 20
Divide 20 by 2:
640 = 2 × 2 × 2 × 2 × 2 × 2 × 10
Divide 10 by 2 (not divisible), so move on to the next prime number, which is 5:
640 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 1
Drop the "× 1" since it doesn't affect the product. Therefore, the prime factorization of 640 is 2⁷ × 5.

Now, let's simplify the fraction:

To reduce the fraction 165/180 to its lowest terms, you need to find the greatest common divisor (GCD) of the two numbers and divide both the numerator and denominator by it.

First, find the prime factorization of both 165 and 180:

165 = 3 × 5 × 11
180 = 2² × 3² × 5

Next, find the common prime factors for both numbers, which are 3 and 5.

Since 3 and 5 are common, find the smallest exponent for each prime factor:
3 appears as 1 time in 165 and 2 times in 180.
5 appears as 1 time in both 165 and 180.

Therefore, the GCD is 3 × 5 = 15.

Finally, divide both the numerator and denominator of the fraction by 15 to simplify it:
165/180 = (165 ÷ 15) / (180 ÷ 15) = 11/12

Thus, the fraction 165/180 simplifies to 11/12 in its lowest terms.