Question

Find the square roots of the following numbers by the Prime Factorisation Method. 

1. 729 
2. 400 
3.  1764 
4.  4096 
5.  7744 
6.  9604 
7.  5929 
8.  9216 
9.  529 
10.  8100

Answer 1

(i)\(729\) can be factorised as follows.

3	729
3	243
3	81
3	27
3	9
3	3
	1

\(729 = \underline{3 × 3} ×\underline{ 3 × 3} × \underline{3 × 3}\)
\(\therefore \sqrt{729}=3\times3\times3=27\)

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(ii) \(400\) can be factorised as follows.

2	400
2	200
2	100
2	50
5	25
5	5
	1

\(400 = \underline{2 × 2} × \underline{2 × 2} × \underline{5 × 5}\)
\(\therefore \sqrt{400}=2\times2\times5=20\)

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(iii) \(1764\) can be factorised as follows.

2	1764
2	882
3	441
3	147
7	49
7	7
	1

1764 = 2 × 2 × 3 × 3 × 7 × 7
\(\therefore \sqrt{1764}=2\times3\times7=42\)

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(iv) \(4096\) can be factorised as follows.

2	256
2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

\(\sqrt{4096}= 2\times 2\times2\times2\times2\times2=64\)

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(v)\( 7744\)

2	7744
2	3872
2	1936
2	968
2	484
2	242
2	121
11	11
11	1

\(\sqrt{7744}   = 2 \times 2 \times 2 \times 11    = 88\)

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(vi) \( 9604  \) 

2	9604
2	4802
7	2401
7	343
7	49
7	7
	1

 \(\sqrt{9604}     \\= 2 \times 7 \times 7    = 98\) 

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(vii)\( 5929\)
 

7	5929
7	847
11	121
11	11
	1

  \(\sqrt{5929} \\= 7 \times 11    = 77\)

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(viii) \(9216\)    

2	9216
2	4608
2	2304
2	1152
2	576
2	288
2	144
2	72
2	36
2	18
3	9
3	3
	1

\(\sqrt{9216}\\= 2 \times 2 \times 2 \times 2 \times2 \times 3    = 96\)

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(ix) \(529\) 

23	529
23	23
	1

\(\sqrt{529}     = 23\)

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(x) \(8100\)

2	8100
2	4050
3	2025
3	675
3	225
3	75
5	25
5	5
	1

 \(\sqrt{8100}     \\= 2 \times 3 \times 3 \times 5    = 90 \)