To determine the simplest form of [tex]\(\sqrt{1,225}\)[/tex], let's break down the number 1225 step by step.
1. Prime Factorization:
First, find the factors of 1225:
- 1225 is divisible by 5 since the last digit is 5.
[tex]\[ 1225 \div 5 = 245 \][/tex]
- Continue factoring 245 by 5.
[tex]\[ 245 \div 5 = 49 \][/tex]
- Finally, factor 49, which is [tex]\(7 \times 7\)[/tex].
So, the prime factorization of 1225 is:
[tex]\[ 1225 = 5 \times 5 \times 7 \times 7 \][/tex]
2. Rewriting the Expression:
Now, we can rewrite the number 1225 in terms of its prime factors:
[tex]\[ 1225 = (5 \times 5) \times (7 \times 7) = 25 \times 49 \][/tex]
3. Taking the Square Root:
To find the square root of 1225, take the square root of each squared term:
[tex]\[ \sqrt{1225} = \sqrt{(5 \times 5) \times (7 \times 7)} = \sqrt{5^2 \times 7^2} \][/tex]
Since the square root and square are inverse operations, they cancel each other out:
[tex]\[ \sqrt{5^2} = 5 \][/tex]
[tex]\[ \sqrt{7^2} = 7 \][/tex]
Thus,
[tex]\[ \sqrt{1225} = 5 \times 7 = 35 \][/tex]
Therefore, the simplest form of [tex]\(\sqrt{1,225}\)[/tex] is [tex]\(\boxed{35}\)[/tex].
