Question

Find the square root of the following using prime factorization.
\[
  {\text{A}}{\text{. 576}} \\
  {\text{B}}{\text{. 1444}} \\
  {\text{C}}{\text{. 2401}} \\
  {\text{D}}{\text{. 3364}} \\
  {\text{E}}{\text{. 4761}} \\
 \]

Answer 1

Hint: When the number is multiplied by itself, it results in the square of a number. Alternatively, when a number is raised to the power of half, it results in the square root of a number. The square root of a number is denoted as \[\sqrt x \], and so, when the result is multiplied by itself, produces \[\sqrt x \times \sqrt x = x\]the original number.
To find the square root of a number using the method of prime factorization first, we will have to find the prime factor of the number, and these numbers are grouped in pairs, which are the same, and then their product is found. For example, the prime factor of \[\left( c \right) = a \times a \times a \times b \times b \times a \times c \times c\]the number which is grouped in pair as \[\left( c \right) = \underline {a \times a} \times \underline {a \times a} \times \underline {b \times b} \times \underline {c \times c} = a \times a \times b \times c\].

Complete step by step answer:
A) Using the prime factorization method to find square root find their prime factors:
\[
   2\underline {\left| {576} \right.} \\
   2\underline {\left| {288} \right.} \\
   2\underline {\left| {144} \right.} \\
   2\underline {\left| {72} \right.} \\
   2\underline {\left| {36} \right.} \\
   2\underline {\left| {18} \right.} \\
   3\underline {\left| 9 \right.} \\
   3 \\
 \]
Hence we can write,\[\left( {576} \right) = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\]
Now make a pair of the same numbers of factors as:
\[\left( {576} \right) = \underline {2 \times 2} \times \underline {2 \times 2} \times \underline {2 \times 2} \times \underline {3 \times 3} = 2 \times 2 \times 2 \times 3 = 24\]
Hence, the square root of \[\sqrt {576} = 24\].

B) Using the prime factorization method to find square root find their prime factors:
\[
   2\underline {\left| {1444} \right.} \\
   2\underline {\left| {722} \right.} \\
   19\underline {\left| {361} \right.} \\
   19 \\
 \]
Hence we can write \[\left( {1444} \right) = 2 \times 2 \times 19 \times 19\]
Now make the pair of same numbers of the factors as:
\[\left( {1444} \right) = \underline {2 \times 2} \times \underline {19 \times 19} = 2 \times 19 = 38\]
Hence, the square root of\[\sqrt {1444} = 38\].

C) Using the prime factorization method to find square root find their prime factors:
\[
   7\underline {\left| {2401} \right.} \\
   7\underline {\left| {343} \right.} \\
   7\underline {\left| {49} \right.} \\
   7 \\
 \]
So we write \[\left( {2401} \right) = 7 \times 7 \times 7 \times 7\]
Now make the pairs of the same numbers as:
\[\left( {2401} \right) = \underline {7 \times 7} \times \underline {7 \times 7} = 7 \times 7 = 49\]
Hence, the square root of \[\sqrt {2401} = 49\].

D) Using the prime factorization method to find square root find their prime factors:
\[
   2\underline {\left| {3364} \right.} \\
   2\underline {\left| {1682} \right.} \\
   19\underline {\left| {841} \right.} \\
   19 \\
 \]
Hence we write \[\left( {3364} \right) = 2 \times 2 \times 29 \times 29\]
Now make the pair of the same number as:
\[\left( {3364} \right) = \underline {2 \times 2} \times \underline {29 \times 29} = 2 \times 29 = 58\]
Hence, the square root of \[\sqrt {3364} = 58\].

E) Using the prime factorization method to find square root find their prime factors:
\[
   3\underline {\left| {4761} \right.} \\
   3\left| {1587} \right. \\
   23\underline {\left| {529} \right.} \\
   23 \\
 \]
Hence we write \[\left( {4761} \right) = 3 \times 3 \times 23 \times 23\]
Now make the pair of the same number as:
\[\left( {4761} \right) = \underline {3 \times 3} \times \underline {23 \times 23} = 3 \times 23 = 69\]
Hence the square root of \[\sqrt {4762} = 69\].

Note: Square root of a number can be produced either by the long division method or by the prime factorization method. The numbers which have only two factors, i.e., 1 and itself, are termed as prime numbers.