# FINDING SQUARE ROOT BY PRIME FACTORIZATION METHOD

The following steps will be useful to find square root of a number by prime factorization.

(i) Decompose the number inside the square root into prime factors.

(ii) Inside the square root, for every two same numbers multiplied, one number can be taken out of the square root.

(iii) Combine the like square root terms using mathematical operations.

Example :

√3 - √12 + √27 = √3 + √(2 ⋅ 2 ⋅ 3) + √(3 ⋅ 3 ⋅ 3)

√3 - 2√3 + 3√3

= 2√3

## Practice Questions

Find the square root each of the following numbers prime factorization.

Question 1 :

36

Decompose 36 into prime factors using synthetic division. √36 = √(2 ⋅ 2 ⋅ 3 ⋅ 3)

(2 ⋅ 3)

= 6

Question 2 :

144

Decompose 144 into prime factors using synthetic division. √144 = √(2 ⋅ ⋅ ⋅ ⋅ 3 ⋅ 3)

(2 ⋅ ⋅ 3)

= 12

Question 3 :

256

Decompose 256 into prime factors using synthetic division. √256 = √(2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2)

(2 ⋅ ⋅ ⋅ 2)

= 16

Question 4 :

324

Decompose 324 into prime factors using synthetic division. √324 = √(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

= 2 ⋅ 3 ⋅ 3

= 18

Question 5 :

400

Decompose 400 into prime factors using synthetic division. √400 = √(2 ⋅ ⋅ ⋅ ⋅ 5 ⋅ 5)

(2 ⋅ ⋅ 5)

= 20

Question 6 :

625

Decompose 625 into prime factors using synthetic division. √625 = √(5 ⋅ 5 ⋅ 5 ⋅ 5)

= 5 ⋅ 5

= 25

Question 7 :

1024

Decompose 1024 into prime factors using synthetic division. √1024 = √(2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2)

(2 ⋅ ⋅ ⋅ ⋅ 2)

= 32

Question 8 :

2025

Decompose 2025 into prime factors using synthetic division. √2025 = √(5 ⋅ 5 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

(5 ⋅ 3 ⋅ 3)

= 45

Question 9 :

3136

Decompose 3136 into prime factors using synthetic division. √3136 = √(2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 7 ⋅ 7)

(2 ⋅ ⋅ ⋅ 7)

= 56

Question 10 :

4096

Decompose 4096 into prime factors using synthetic division. √4096 = √(2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 2)

(2 ⋅ ⋅ ⋅ ⋅ ⋅ 2)

= 64

Question 11 :

√8 + √125

 √8 = √(2 ⋅ 2 ⋅ 2)√8 = 2 √125 = √(5 ⋅ 5 ⋅ 5)√125 = 5

√8 + √125 = 2 + 5

= 7

Question 12 :

(√34)(√136)

(√34)(√136) = √(34 ⋅ 136)

√(2 ⋅ 17 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 17)

= 2 ⋅ 2 ⋅ 17

= 68

Question 13 :

(14√117) ÷ (7√52)

Decompose 117 and 52 into prime factors using synthetic division. √117 = √(3 ⋅ 3 ⋅ 13)√117 = 3√13 √52 = √(2 ⋅ 2 ⋅ 13)√52 = 2√13

(14√117) ÷ (7√52) :

= 14(3√13) ÷ 7(2√13)

= 42√13 ÷ 14√13

= 42√13/14√13

= 3

Question 14 :

(4√7)2

(4√7)= 4√7 ⋅ 4√7

= (4 ⋅ 4)(√7 ⋅ √7)

= (16)(7)

= 112

Question 15 :

(√5)3(√125)

(√5)3 + √125 = (√5 ⋅ √5  √5)√(5 ⋅ 5 ⋅ 5)

= (5√5)(5√5)

= (5 ⋅ 5)(√5 ⋅ √5)

= (25)(5)

= 125 Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

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