question_answer 12)

Express each of the following as a product of prime factors only in exponential form: (i) \[108\times 192\]                       (ii) \[\text{27}0\]                                              (iii) \[729\times 64\]                       (iv) \[768\]

Answer:

                (i) \[\mathbf{108\times 192}\]                 \[108\times 192\]\[=(2\times 2\times 3\times 3\times 3)\times (2\times 2\times 2\times 2\times 2\times 2\times 3)\] \[=2\times 2\times 3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2\times 3\] \[=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3\] \[={{2}^{8}}\times {{3}^{4}}\]

2	108
2	54
3	27
3	9
3	3
	1

                               

2	192
2	96
2	48
2	24
2	12
2	6
3	3
		1

                it is the required prime factor product form                 (ii) 270                

2	270
3	135
3	45
3	15
5	5
	1

\[\therefore \]  \[270=2\times 3\times 3\times 3\times 5={{2}^{1}}\times {{3}^{3}}\times {{5}^{1}}\]                 It is the required prime factor product form.                 (iii) \[\mathbf{729\times 64}\]                 \[729\times 64=3\times 3\times 3\times 3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2\]                

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

               

2	64
2	32
2	16
2	8
2	4
2	2
	1

                \[={{3}^{6}}\times {{2}^{6}}\]                 It is the required prime factor product form.                 (iv) 768                

2	768
2	384
2	192
2	96
2	48
2	24
2	12
2	6
3	3
	1

\[\therefore \]  \[768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\] \[={{2}^{8}}\times {{3}^{1}}\]     It is the required prime factor product form.