Certainly! Let's solve each part of the question step-by-step:
### a. Find the square root of 285156 using prime factorization.
1. Prime Factorization of 285156:
The number 285156 can be broken down into its prime factors as follows:
[tex]\[
285156 = 2^2 \times 3^2 \times 89^2
\][/tex]
2. Square Root Calculation:
- To find the square root from the prime factorization, take the square root of each distinct prime factor raised to half its original power:
[tex]\[
\sqrt{285156} = \sqrt{2^2 \times 3^2 \times 89^2} = 2^1 \times 3^1 \times 89^1 = 2 \times 3 \times 89 = 534
\][/tex]
Hence, the square root of 285156 is 534.
### b. Find the square root of 28224 using prime factorization.
1. Prime Factorization of 28224:
The number 28224 can be broken down into its prime factors as follows:
[tex]\[
28224 = 2^6 \times 3^2 \times 7^2
\][/tex]
2. Square Root Calculation:
- To find the square root from the prime factorization, take the square root of each distinct prime factor raised to half its original power:
[tex]\[
\sqrt{28224} = \sqrt{2^6 \times 3^2 \times 7^2} = 2^3 \times 3^1 \times 7^1 = 8 \times 3 \times 7 = 168
\][/tex]
Hence, the square root of 28224 is 168.
### c. Find the square root of 2097.64.
- Since 2097.64 is a decimal number, we can find its square root directly:
[tex]\[
\sqrt{2097.64} = 45.8
\][/tex]
Therefore, the square root of 2097.64 is 45.8.
### d. Find the square root of [tex]\(\frac{1156}{1296}\)[/tex].
1. Square Root Calculation:
- First, find the square root of the numerator (1156) and the denominator (1296) separately:
[tex]\[
\sqrt{1156} = 34
\][/tex]
[tex]\[
\sqrt{1296} = 36
\][/tex]
2. Square Root of the Fraction:
- Now, find the square root of the fraction by dividing the square root of the numerator by the square root of the denominator:
[tex]\[
\sqrt{\frac{1156}{1296}} = \frac{\sqrt{1156}}{\sqrt{1296}} = \frac{34}{36} = \frac{17}{18}
\][/tex]
Therefore, the square root of [tex]\(\frac{1156}{1296}\)[/tex] is approximately 0.9444444444444444 when expressed as a decimal.
In summary, the answers are:
a. [tex]\( \sqrt{285156} = 534 \)[/tex]
b. [tex]\( \sqrt{28224} = 168 \)[/tex]
c. [tex]\( \sqrt{2097.64} = 45.8 \)[/tex]
d. [tex]\( \sqrt{\frac{1156}{1296}} \approx 0.9444444444444444 \)[/tex]
