To find the simplest form of [tex]\(\sqrt{1,225}\)[/tex], we can proceed step by step.
1. Identify the number: We start with the number [tex]\(1,225\)[/tex].
2. Prime Factorization: We break [tex]\(1,225\)[/tex] down into its prime factors. For that, let's first recognize if [tex]\(1,225\)[/tex] is a perfect square of some integer [tex]\(n\)[/tex].
3. Check for smallest factors:
- Start with the smallest prime number [tex]\(2\)[/tex]. [tex]\(1,225\)[/tex] is odd, so it is not divisible by [tex]\(2\)[/tex].
- Next, try [tex]\(3\)[/tex]. Sum of digits of [tex]\(1,225\)[/tex] is [tex]\(1 + 2 + 2 + 5 = 10\)[/tex]. Since [tex]\(10\)[/tex] is not divisible by [tex]\(3\)[/tex], [tex]\(1,225\)[/tex] is not divisible by [tex]\(3\)[/tex].
- Next, try [tex]\(5\)[/tex]. The last digit of [tex]\(1,225\)[/tex] is [tex]\(5\)[/tex], so [tex]\(1,225\)[/tex] is divisible by [tex]\(5\)[/tex]. Dividing, we get [tex]\(1,225 \div 5 = 245\)[/tex].
4. Continue factorizing [tex]\(245\)[/tex]:
- [tex]\(245\)[/tex] ends in [tex]\(5\)[/tex], so it is again divisible by [tex]\(5\)[/tex]. Dividing, we get [tex]\(245 \div 5 = 49\)[/tex].
5. Factorize [tex]\(49\)[/tex]:
- Recognize that [tex]\(49\)[/tex] is [tex]\(7 \times 7\)[/tex].
6. Prime factors: Combining these steps, we have:
[tex]\[
1,225 = 5 \times 5 \times 7 \times 7
\][/tex]
7. Simplify the square root: The formula for simplifying a square root calculation for a composite number [tex]\(n\)[/tex] given its prime factors is:
[tex]\[
\sqrt{n} = \sqrt{(a^2 \times b^2 \times ...)} = a \times b \times ...
\][/tex]
Applying this to our factors, we get:
[tex]\[
\sqrt{1,225} = \sqrt{(5 \times 5) \times (7 \times 7)} = 5 \times 7 = 35
\][/tex]
Therefore, the simplest form of [tex]\(\sqrt{1,225}\)[/tex] is:
[tex]\[
\boxed{35}
\][/tex]
