Sure, let's solve these two problems step-by-step.
### (1) Finding [tex]\(\sqrt{36864}\)[/tex]:
We need to determine the square root of 36864.
The square root of 36864 is calculated as follows:
[tex]\[
\sqrt{36864} = 192.0
\][/tex]
So, [tex]\(\sqrt{36864} = 192.0\)[/tex].
### (2) Finding [tex]\(\sqrt{2401}\)[/tex] by prime factorization:
To find [tex]\(\sqrt{2401}\)[/tex] using prime factorization:
1. First, we factorize 2401 into its prime factors.
2. Upon factorizing, we see:
[tex]\[
2401 = 7 \times 7 \times 7 \times 7 = 7^4
\][/tex]
3. Thus, the square root of 2401 can be written as:
[tex]\[
\sqrt{2401} = \sqrt{7^4}
\][/tex]
4. Since [tex]\(\sqrt{a^2} = a\)[/tex], we get:
[tex]\[
\sqrt{7^4} = 7^2 = 49.0
\][/tex]
So, [tex]\(\sqrt{2401} = 49.0\)[/tex].
Thus, the answers to the questions are:
1. [tex]\(\sqrt{36864} = 192.0\)[/tex]
2. [tex]\(\sqrt{2401} = 49.0\)[/tex]
