To solve the problem of finding the simplest form of [tex]\(\sqrt{3025}\)[/tex], we need to follow a step-by-step approach. Let's dive into each step carefully.
Step 1: Find the prime factors of 3025
First, identify the factors of 3025.
1. 3025 is divisible by 5:
[tex]\[
3025 \div 5 = 605
\][/tex]
2. 605 is also divisible by 5:
[tex]\[
605 \div 5 = 121
\][/tex]
3. 121 is divisible by 11:
[tex]\[
121 \div 11 = 11
\][/tex]
4. 11 is also divisible by 11:
[tex]\[
11 \div 11 = 1
\][/tex]
Thus, the prime factors of 3025 are:
[tex]\[
3025 = 5 \times 5 \times 11 \times 11 = 5^2 \times 11^2
\][/tex]
Step 2: Simplify the square root using the prime factors
Now, we need to take the square root of 3025. We use the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex].
So, we break it down:
[tex]\[
\sqrt{3025} = \sqrt{5^2 \times 11^2}
\][/tex]
Next, we separate the square root:
[tex]\[
\sqrt{3025} = \sqrt{5^2} \times \sqrt{11^2}
\][/tex]
Since [tex]\(\sqrt{5^2} = 5\)[/tex] and [tex]\(\sqrt{11^2} = 11\)[/tex], we multiply these results:
[tex]\[
\sqrt{3025} = 5 \times 11 = 55
\][/tex]
Conclusion
Therefore, the simplest form of [tex]\(\sqrt{3025}\)[/tex] is [tex]\(55\)[/tex].
Among the provided options:
- 16
- 55
- [tex]\(5^2 \left(11^2\right)\)[/tex]
- [tex]\(5^2 + \left(11^2\right)\)[/tex]
The correct choice is:
[tex]\[
\boxed{55}
\][/tex]
