Sure! Let's follow the steps to find the square root of 2028 by using the prime factorization method.
### Step 1: Prime Factorization
First, we need to find the prime factors of 2028. The prime factorization involves expressing 2028 as a product of prime numbers.
After factorizing 2028, we get:
[tex]\[ 2028 = 2 \times 2 \times 3 \times 13 \times 13 \][/tex]
So, the prime factors of 2028 are:
[tex]\[ 2, 2, 3, 13, 13 \][/tex]
### Step 2: Group the Factors into Pairs
Next, we need to group these factors into pairs to find if they can form perfect squares.
From our list of factors, we can pair them as:
[tex]\[ (2, 2) \text{ and } (13, 13) \][/tex]
We still have one unpaired factor left:
[tex]\[ 3 \][/tex]
### Step 3: Calculate the Product of Paired Factors
To find the square root, we multiply one number from each pair we formed:
[tex]\[ \sqrt{2028} = 2 \times 13 \][/tex]
Calculating this, we get:
[tex]\[ 2 \times 13 = 26 \][/tex]
### Step 4: Conclude the Solution
Thus, by the prime factorization method, the square root of 2028 can be determined to be 26, assuming that we are ignoring any unpaired factors which cannot form part of a perfect square.
So, our steps show:
- Prime Factors: [tex]\([2, 2, 3, 13, 13]\)[/tex]
- Paired Factors: [tex]\([2, 13]\)[/tex]
- Unpaired Factor: [tex]\([3]\)[/tex]
- Resultant Square Root: [tex]\(26\)[/tex]
The square root of 2028, by prime factorization method, is:
[tex]\[ \sqrt{2028} = 26 \][/tex]
