Question

Find the square root of the following by the prime factorization method.
1) 144
2) 1024
3) 3025
4) 9604
5) 15876

Answer 1

 Hint: The factors of a number are the divisor which will completely divide the number without leaving any remainder. To find the square root of a number by prime factorization method, first divide the given number to its prime factor then make the pair of similar factor, so that factor in each pair are equal then square root all the pair of factor and find the product of factor. The square root of a number is a number whose square is equal to the original number.

Complete step-by-step answer:
1) The factors of 144 are \[2 \times 2 \times 2 \times 2 \times 3 \times 3\]
This can also be written as \[144 = {2^2} \times {2^2} \times {3^2}\] arranged in a pair of similar factors.
Now, put square root on both sides-
\[
  \sqrt {144} = \sqrt {{2^2} \times {2^2} \times {3^2}} \\
  \sqrt {144} = 2 \times 2 \times 3 \\
  \sqrt {144} = 12 \\
 \]
Hence, the square root of 144 is 12.

2) The factors of 1024 are \[2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\]
This can also be written as \[1024 = {2^2} \times {2^2} \times {2^2} \times {2^2} \times {2^2}\]
Now, put square root on both sides
\[
  \sqrt {1024} = \sqrt {{2^2} \times {2^2} \times {2^2} \times {2^2} \times {2^2}} \\
  \sqrt {1024} = \,2 \times 2 \times 2 \times 2 \times 2 \\
  \sqrt {1024} = 32 \\
 \]
Hence, the square root of 1024 is 32.

3) The factors of 3025 are \[5 \times 5 \times 11 \times 11\]
This can also be written as \[3025 = {5^2} \times {11^2}\]
Now, put square root on both sides
\[
  \sqrt {3025} = \sqrt {{5^2} \times {{11}^2}} \\
  \sqrt {3025} = 5 \times 11 \\
  \sqrt {3025} = 55 \\
 \]
Hence, the square root of 3025 is 55.

4) The factors of 9604 are \[2 \times 2 \times 7 \times 7 \times 7 \times 7\]
This can also be written as \[9604 = {2^2} \times {7^2} \times {7^2}\]
Now, put square root on both sides
\[
  \sqrt {9604} = \sqrt {{2^2} \times {7^2} \times {7^2}} \\
  \sqrt {9604} = 2 \times 7 \times 7 \\
  \sqrt {9604} = 96 \\
 \]
Hence, the square root of 9604 is 96.

5) The factors of 15876 are \[2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 7 \times 7\]
This can also be written as \[15876 = {2^2} \times {3^4} \times {7^2}\]
Now, put square root on both sides
\[
  \sqrt {15876} = \sqrt {{2^2} \times {3^4} \times {7^2}} \\
  \sqrt {15876} = 2 \times {3^2} \times 7 \\
  \sqrt {15876} = 2 \times 9 \times 7 \\
  \sqrt {15876} = 126 \\
 \]
Hence, the square root of 15876 is 126.

Note: Always find the prime factor of the number and group them into the pairs of similar factors.