Introduction For Prime Power Factorization:
Prime factorization is the method of expressing a number as the product of powers of prime
Example : Let us consider the number 162.
162 can be written as 81 x 2
81 is not a prime number and can be written as
This is called prime power factorization. We have expressed the number as a power of prime numbers
81 = 9 x 9
162 = 9 X 9 X 2
But 9 is not a prime number and can be written as 3 x 3
so 81 = 3 x 3 x 3 x 3
Hence 162 = 2 x 3 x 3 x 3 x 3
This can be expressed in powers of 2 and 3 ss below
81 = `3^(4)`
162 = `2^(1)` x `3^(4)`
Thus we have expressed 162 as a product of powers of prime numbers 2 and 3. This is called prime power factorization.
Steps for Finding Prime Power Factorization:
Step 1: Its is like normal prime factor factorization, first find possibilities factors for given numbers.The process of finding power factors is called prime power factorization.
example: 27
27=3*9
27=3*3*3 prime factors =3*3*3
Step 2: After finding factors we have to arrange the numbers with powers.
Example prime factors of 140
140=2*2*5*7
=22*5*7
Step 3: Then arranging multiply the power factors and the power factors of given numbers is found.
Example prime numbers 625
625=5*5*5*5 .
=54 are prime power factorization
Example Problems for Prime Power Factorization:
Example 1: Find prime power factors for 100.
Solution:
100÷2 = 50
50÷2 = 25
25÷5 = 5
5÷5=1
the prime factors are 2, 2, 5, 5
then prime power factors of 100= 2²*5².
Example 2: Find prime power factors for 48.
Solution: Starting with 2
48 = 24 * 2
= 12 * 2 * 2
= 6 * 2 * 2 * 2
= 3 * 2*2*2*2
prime number 3 so that's as far as we can go:
48 = 3 * 2*2*2*2
prime power factors of 48 = 24 * 32
Example 3: Find the prime power factors of 540.
Solution:
540÷2=2*270
270÷2=2*135
135÷3=3*45
45÷3=3*15
15÷3=3*5
so, the prime power factors of 2*2*3*3*3*5 =22*33*5
Example 4: Find the prime power factors of 125.
Solution:
125÷5=5*25
25÷5=5*5
so, the prime power factors of 5*5*5 =53
Example 5: Find the prime power factors of 220.
Solution:
220÷2=2*110
110÷2=2*55
55÷5=5*11
so, the prime power factors of 2*2*5*11 =22*5*11
My Previous Blog :- http://learnmathproblems.blogspot.in/2012/10/logarithm-of-complex-number.html
Prime factorization is the method of expressing a number as the product of powers of prime
Example : Let us consider the number 162.
162 can be written as 81 x 2
81 is not a prime number and can be written as
This is called prime power factorization. We have expressed the number as a power of prime numbers
81 = 9 x 9
162 = 9 X 9 X 2
But 9 is not a prime number and can be written as 3 x 3
so 81 = 3 x 3 x 3 x 3
Hence 162 = 2 x 3 x 3 x 3 x 3
This can be expressed in powers of 2 and 3 ss below
81 = `3^(4)`
162 = `2^(1)` x `3^(4)`
Thus we have expressed 162 as a product of powers of prime numbers 2 and 3. This is called prime power factorization.
Steps for Finding Prime Power Factorization:
Step 1: Its is like normal prime factor factorization, first find possibilities factors for given numbers.The process of finding power factors is called prime power factorization.
example: 27
27=3*9
27=3*3*3 prime factors =3*3*3
Step 2: After finding factors we have to arrange the numbers with powers.
Example prime factors of 140
140=2*2*5*7
=22*5*7
Step 3: Then arranging multiply the power factors and the power factors of given numbers is found.
Example prime numbers 625
625=5*5*5*5 .
=54 are prime power factorization
Example Problems for Prime Power Factorization:
Example 1: Find prime power factors for 100.
Solution:
100÷2 = 50
50÷2 = 25
25÷5 = 5
5÷5=1
the prime factors are 2, 2, 5, 5
then prime power factors of 100= 2²*5².
Example 2: Find prime power factors for 48.
Solution: Starting with 2
48 = 24 * 2
= 12 * 2 * 2
= 6 * 2 * 2 * 2
= 3 * 2*2*2*2
prime number 3 so that's as far as we can go:
48 = 3 * 2*2*2*2
prime power factors of 48 = 24 * 32
Example 3: Find the prime power factors of 540.
Solution:
540÷2=2*270
270÷2=2*135
135÷3=3*45
45÷3=3*15
15÷3=3*5
so, the prime power factors of 2*2*3*3*3*5 =22*33*5
Example 4: Find the prime power factors of 125.
Solution:
125÷5=5*25
25÷5=5*5
so, the prime power factors of 5*5*5 =53
Example 5: Find the prime power factors of 220.
Solution:
220÷2=2*110
110÷2=2*55
55÷5=5*11
so, the prime power factors of 2*2*5*11 =22*5*11
My Previous Blog :- http://learnmathproblems.blogspot.in/2012/10/logarithm-of-complex-number.html
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