Find the prime factorization of 33^5*39*25^12
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11^5 * 3^5 * 3^1 * 13^1 * (5^2)^12
3^6 * 5^24 * 11 * 13
To find the prime factorization of 33^5 * 39 * 25^12, we need to break down each number into its prime factors.
Starting with 33^5:
33 is divisible by 3. Therefore, we can write 33 as 3 * 11.
33^5 = (3 * 11)^5 = 3^5 * 11^5 = 243 * 161051 = 39,169,053.
Next, let's look at 39:
39 is divisible by 3 and 13. Therefore, we can write 39 as 3 * 13.
So far, we have:
33^5 * 39 = 39,169,053 * (3 * 13) = 39,169,053 * 39 = 1,528,091,967.
Now, let's break down 25^12:
25 is divisible by 5. Therefore, we can write 25 as 5 * 5.
So, 25^12 = (5 * 5)^12 = 5^12 * 5^12 = 244,140,625.
Finally, we can multiply everything together:
33^5 * 39 * 25^12 = 1,528,091,967 * 244,140,625 = 372,200,339,419,921,875.
Therefore, the prime factorization of 33^5 * 39 * 25^12 is:
2^4 * 3^5 * 5^14 * 7^0 * 11^5 * 13^1 * 17^0 * 19^0 * 23^0 * 29^0 * 31^0 * 37^0 * 41^0 * 43^0 * 47^0 * 53^0 * 59^0 * 61^0 * 67^0 * 71^0 * 73^0 * 79^0 * 83^0 * 89^0 * 97^0 * 101^0 * 103^0 * 107^0 * 109^0 * 113^0 * 127^0 * 131^0 * 137^0 * 139^0 * 149^0 * 151^0 * 157^0 * 163^0 * 167^0 * 173^0 * 179^0 * 181^0 * 191^0 * 193^0 * 197^0 * 199^0 * 211^0 * 223^0 * 227^0 * 229^0 * 233^0 * 239^0 * 241^0 * 251^0 * 257^0 * 263^0 * 269^0 * 271^0 * 277^0 * 281^0 * 283^0 * 293^0 * 307^0 * 311^0 * 313^0 * 317^0 * 331^0 * 337^0 * 347^0 * 349^0 * 353^0 * 359^0 * 367^0 * 373^0 * 379^0 * 383^0 * 389^0 * 397^0 * 401^0 * 409^0 * 419^0 * 421^0 * 431^0 * 433^0 * 439^0 * 443^0 * 449^0 * 457^0 * 461^0 * 463^0 * 467^0 * 479^0 * 487^0 * 491^0 * 499^0 * 503^0 * 509^0 * 521^0 * 523^0 * 541^0 * 547^0 * 557^0 * 563^0 * 569^0 * 571^0 * 577^0 * 587^0 * 593^0 * 599^0 * 601^0 * 607^0 * 613^0 * 617^0 * 619^0 * 631^0 * 641^0 * 643^0 * 647^0 * 653^0 * 659^0 * 661^0 * 673^0 * 677^0 * 683^0 * 691^0 * 701^0 * 709^0 * 719^0 * 727^0 * 733^0 * 739^0 * 743^0 * 751^0 * 757^0 * 761^0 * 769^0 * 773^0 * 787^0 * 797^0 * 809^0 * 811^0 * 821^0 * 823^0 * 827^0 * 829^0 * 839^0 * 853^0 * 857^0 * 859^0 * 863^0 * 877^0 * 881^0 * 883^0 * 887^0 * 907^0 * 911^0 * 919^0 * 929^0 * 937^0 * 941^0 * 947^0 * 953^0 * 967^0 * 971^0 * 977^0 * 983^0 * 991^0 * 997^0.
To find the prime factorization of a given number, we need to express it as a product of its prime factors.
Let's break down the given expression:
33^5 * 39 * 25^12
First, let's simplify each term one by one.
Prime factorization of 33:
33 can be written as 3 * 11, so its prime factorization is 3 * 11.
Prime factorization of 39:
39 can be written as 3 * 13, so its prime factorization is 3 * 13.
Prime factorization of 25:
25 can be written as 5 * 5, so its prime factorization is 5 * 5.
Now, let's simplify the entire expression:
(3 * 11)^5 * (3 * 13) * (5 * 5)^12
Next, we can apply the exponent to each term:
3^5 * 11^5 * 3 * 13 * 5^12 * 5^12
Now, we can rearrange the terms:
3^5 * 3 * 13 * 11^5 * 5^12 * 5^12
Finally, we can simplify further:
3^6 * 5^24 * 11^5 * 13
So, the prime factorization of 33^5 * 39 * 25^12 is 3^6 * 5^24 * 11^5 * 13.