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Finding Square Root of a Perfect Square by Prime Factorization

Square Root Of A Perfect Square By Prime Factorization :

In order to find the square root of a perfect square by prime factorization, follow the following steps.
Step I– Obtain the given number.
Step II– Resolve the given number into prime factors by successive division.
Step III– Make pairs of prime factors such that both the factors in each pair are equal. Since the number is a perfect square, you will be able to make an exact number of pairs of prime factors.
Step IV– Take one factor from each pair.
Step V– Find the product of factors obtained in step IV.
Step VI– The product obtained in step V is the required square root.

Sum and products of roots calculator.

Illustrative Examples :

Square-Root-of-a-Perfect-Square-by-Prime-Factorization-example-1Example 1 : Find the square root of 576.
Solution : By prime factorization,
we get 576=(2 x 2) x (2 x 2) x (2 x 2) x (3 x 3)                            [ Make pairs of factors]
\( \therefore \)   \( \sqrt { 576 } \) = 2x2x2x3 = 24     [ Pickup one factor from each pair ]

 

 

 

Square-Root-of-a-Perfect-Square-by-Prime-Factorization-example-2 Example 2: Find the square root of \( \sqrt { 7744 } \)
Solution: Step 1. Split the given number into prime factors.
7744 =2x2x2x2x2x2x11x11

  Step 2. Form pairs of like factors.
7744 =(2×2)x(2×2)x(2×2)x(11×11)

 

 

Step 3. From each pair. pick out one prime factor

1st pair              2nd pair           3rd pair            4th pair
                              (2×2)       x            (2×2)          x     (2×2)           x      (11×11)
Pick out                Pick out              Pick out               Pick out
one 2                    one 2                    one 2                     one 2

Step 4. Multiply the factors so picked.

The product is the square root of the given number.
\( \sqrt { 7744} \) =2x2x2x11 = 88.

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