Prime Factorisation

Step 1: The aim of prime factorisation is to write the number using prime bases only. Step 2: If a base like (45) or (121) remains, it is composite and must be broken further. Step 3: Before writing the final answer, check whether every base is prime.

Step 1: Write (680400=16\times42525). Step 2: (16=2^4) and (42525=3^5\times5^2\times7), so the correct form is (2^4\times3^5\times5^2\times7). Step 3: Do not leave 42525 in the final form.

Step 1: Write (627264=64\times9801). Step 2: (64=2^6) and (9801=3^4\times11^2), so (627264=2^6\times3^4\times11^2). Step 3: Convert 9801 into prime powers.

Step 1: Write (379080=8\times47385). Step 2: (8=2^3) and (47385=3^6\times5\times13), so the correct form is (2^3\times3^6\times5\times13). Step 3: Give 47385 its complete prime form.

Step 1: Write (196000=32\times6125). Step 2: (32=2^5) and (6125=5^3\times7^2), so (196000=2^5\times5^3\times7^2). Step 3: Do not keep 6125 in the final answer.

Step 1: Write (289575=81\times3575). Step 2: (81=3^4) and (3575=5^2\times11\times13), so (289575=3^4\times5^2\times11\times13). Step 3: Give 3575 its complete prime form too.

Step 1: Write (177408=256\times693). Step 2: (256=2^8) and (693=3^2\times7\times11), so the correct form is (2^8\times3^2\times7\times11). Step 3: Convert 693 into prime form.

Step 1: Write (529200=16\times33075). Step 2: (33075=3^3\times5^2\times7^2), so (529200=2^4\times3^3\times5^2\times7^2). Step 3: Do not leave 33075 in the final form.

Step 1: Write (155520=128\times1215). Step 2: (128=2^7) and (1215=3^5\times5), so (155520=2^7\times3^5\times5). Step 3: Convert 1215 into prime powers.

Step 1: Write (174636=4\times43659). Step 2: (4=2^2) and (43659=3^4\times7^2\times11), so the correct form is (2^2\times3^4\times7^2\times11). Step 3: Convert 43659 into prime powers.

Step 1: Write (79200=32\times2475). Step 2: (2475=3^2\times5^2\times11), so (79200=2^5\times3^2\times5^2\times11). Step 3: Give 2475 its complete prime form.

Step 1: Write (415800=8\times51975). Step 2: (51975=3^3\times5^2\times7\times11), so (415800=2^3\times3^3\times5^2\times7\times11). Step 3: Do not leave 51975 in the final form.

Step 1: Write (762048=64\times11907). Step 2: (64=2^6) and (11907=3^5\times7^2), so (762048=2^6\times3^5\times7^2). Step 3: Write 11907 as prime powers.

Step 1: Write (137200=16\times8575). Step 2: (16=2^4) and (8575=5^2\times7^3), so the correct form is (2^4\times5^2\times7^3). Step 3: Give 8575 its final prime form.

Step 1: Write (127575=729\times175). Step 2: (729=3^6) and (175=5^2\times7), so (127575=3^6\times5^2\times7). Step 3: Give prime form to both 729 and 175.

Step 1: Write (453600=32\times14175). Step 2: (14175=3^4\times5^2\times7), so (453600=2^5\times3^4\times5^2\times7). Step 3: Do not leave 14175 in the final form.

Step 1: Write (139392=128\times1089). Step 2: (128=2^7) and (1089=3^2\times11^2), so (139392=2^7\times3^2\times11^2). Step 3: Convert 1089 into prime powers.

Step 1: Write (117000=8\times14625). Step 2: (14625=3^2\times5^3\times13), so (117000=2^3\times3^2\times5^3\times13). Step 3: Give 14625 its complete prime form.

Step 1: Write (475200=64\times7425). Step 2: (7425=3^3\times5^2\times11), so (475200=2^6\times3^3\times5^2\times11). Step 3: Do not leave 7425 in the final form.

Step 1: Write (99225=81\times1225). Step 2: (81=3^4) and (1225=5^2\times7^2), so (99225=3^4\times5^2\times7^2). Step 3: Convert 1225 into prime powers.

Step 1: Write (518400=256\times2025). Step 2: (256=2^8) and (2025=3^4\times5^2), so (518400=2^8\times3^4\times5^2). Step 3: Give 2025 its final prime form.

Step 1: (529200=16\times33075). Step 2: (33075=3^3\times5^2\times7^2), so the power of 7 is 2. Step 3: Comparing gives (q=2).

Step 1: Write (627264=64\times9801). Step 2: (64=2^6) and (9801=3^4\times11^2), so the power of 2 is 6. Step 3: From the given form, (a=6).

Step 1: (680400=16\times42525). Step 2: (42525=3^5\times5^2\times7), so the power of 3 is 5. Step 3: Comparing gives (b=5).

Step 1: Write (137200=16\times8575). Step 2: (8575=5^2\times7^3), so the power of 7 is 3. Step 3: From the given form, (p=3).