Alright, let's break down the problem step-by-step.
### Step 1: Evaluate the Expression [tex]\( 120 - 2^1 \times 3 \times 5 \)[/tex]
First, we solve the expression:
[tex]\[ 120 - 2^1 \times 3 \times 5 \][/tex]
Calculate [tex]\( 2^1 \)[/tex]:
[tex]\[ 2^1 = 2 \][/tex]
Multiply the results:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
Subtract from 120:
[tex]\[ 120 - 30 = 90 \][/tex]
So the evaluated expression is:
[tex]\[ 120 - 2^1 \times 3 \times 5 = 90 \][/tex]
### Step 2: Compute [tex]\( n = 120 \times 108 \)[/tex]
Next, calculate the product:
[tex]\[ 120 \times 108 = 12960 \][/tex]
### Step 3: Prime Factorization of 120 and 108
Now, we factorize both 120 and 108 into their prime factors.
#### Prime Factorization of 120:
[tex]\[ 120 = 2 \times 60 \][/tex]
[tex]\[ 60 = 2 \times 30 \][/tex]
[tex]\[ 30 = 2 \times 15 \][/tex]
[tex]\[ 15 = 3 \times 5 \][/tex]
Therefore:
[tex]\[ 120 = 2^3 \times 3 \times 5 \][/tex]
#### Prime Factorization of 108:
[tex]\[ 108 = 2 \times 54 \][/tex]
[tex]\[ 54 = 2 \times 27 \][/tex]
[tex]\[ 27 = 3 \times 9 \][/tex]
[tex]\[ 9 = 3 \times 3 \][/tex]
Therefore:
[tex]\[ 108 = 2^2 \times 3^3 \][/tex]
### Step 4: Combine the Prime Factors
Combine the prime factors from [tex]\( 120 = 2^3 \times 3 \times 5 \)[/tex] and [tex]\( 108 = 2^2 \times 3^3 \)[/tex].
Sum the exponents for each prime:
- For the prime number [tex]\( 2 \)[/tex]:
[tex]\[ 2^3 \times 2^2 = 2^{3+2} = 2^5 \][/tex]
- For the prime number [tex]\( 3 \)[/tex]:
[tex]\[ 3 \times 3^3 = 3^{1+3} = 3^4 \][/tex]
- For the prime number [tex]\( 5 \)[/tex]:
There's no [tex]\( 5 \)[/tex] in 108, so the power remains the same:
[tex]\[ 5^1 \][/tex]
### Step 5: Write [tex]\( n \)[/tex] as a Product of Prime Factors
Combining these, we get:
[tex]\[ 120 \times 108 = 2^5 \times 3^4 \times 5^1 \][/tex]
Hence, [tex]\( n = 12960 \)[/tex] can be written as a product of powers of its prime factors:
[tex]\[ n = 2^5 \times 3^4 \times 5^1 \][/tex]
To recap, the final answers are:
- The evaluated expression [tex]\( 120 - 2^1 \times 3 \times 5 \)[/tex] is [tex]\( 90 \)[/tex].
- The product [tex]\( n \)[/tex] is [tex]\( 120 \times 108 = 12960 \)[/tex].
- The prime factorization of [tex]\( n = 12960 \)[/tex] is [tex]\( 2^5 \times 3^4 \times 5^1 \)[/tex].
