Prime Factorisation

Step 1: Write (1800=18\times100). Step 2: (18=2\times3^2) and (100=2^2\times5^2), so (1800=2^3\times3^2\times5^2). Step 3: Break 18 and 100 separately into prime form.

Step 1: Write (2025=81\times25). Step 2: (81=3^4) and (25=5^2), so (2025=3^4\times5^2). Step 3: Convert 81 and 25 into prime powers.

Step 1: Divide 2187 repeatedly by 3. Step 2: Seven factors of 3 give (2187=3^7). Step 3: 9 and 27 are composite, so keep base 3 in the final form.

Step 1: (2401) can be written as (49\times49). Step 2: Since (49=7^2), (2401=7^4). Step 3: 49 is composite, so write the power of 7 in final prime form.

Step 1: Write (2500=25\times100). Step 2: (25=5^2) and (100=2^2\times5^2), so (2500=2^2\times5^4). Step 3: The total power of 5 is 4, so count carefully.

Step 1: The prime factorisation of 600 is (2^3\times3\times5^2). Step 2: In the given form, the power of 2 is (a). Step 3: Comparing gives (a=3).

Step 1: (675=27\times25). Step 2: (27=3^3) and (25=5^2), so (675=3^3\times5^2). Step 3: Comparing gives (a=3).

Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: Evaluating powers first makes calculation easier.

Step 1: Calculate (3^4=81). Step 2: (2\times81=162). Step 3: Simplify the factor with a power first.

Step 1: (2^3=8). Step 2: (8\times3\times7=168). Step 3: Multiply to get the number from prime factorisation.

Step 1: (3^2=9) and (5^2=25). Step 2: (9\times25=225). Step 3: Simplify both powers first.

Step 1: In final prime factorisation, every factor must be prime. Step 2: 2, 3, 5, 7, and 11 are prime. Step 3: 6, 21, 35, and 33 are composite, so they cannot remain in final form.

Step 1: Final prime factorisation must not contain a composite factor. Step 2: 10 is composite, so (10\times3^2\times7) is not final form. Step 3: Change 10 into (2\times5).

Step 1: Divide 2048 repeatedly by 2. Step 2: Eleven factors of 2 give (2048=2^{11}). Step 3: 4, 32, and 64 are composite, so write the power of 2 in final form.

Step 1: Divide 729 repeatedly by 3. Step 2: Six factors of 3 give (729=3^6). Step 3: 9 and 27 are composite, so keep prime base 3.

Step 1: (361=19\times19). Step 2: Since 19 is prime, (361=19^2). Step 3: In a square number, the same prime appears twice.

Step 1: Write (1331=11\times121). Step 2: (121=11^2), so (1331=11^3). Step 3: 121 is composite, so write (11^3) in the final form.

Step 1: Divide 4096 repeatedly by 2. Step 2: Twelve factors of 2 give (4096=2^{12}). Step 3: 64 and 16 are composite, so they are not final prime forms.

Step 1: Write (3125=5\times625). Step 2: Since (625=5^4), (3125=5^5). Step 3: 25 and 125 are composite forms, so write a power of 5 in the final answer.

Step 1: Recognise (2744=14^3). Step 2: Since (14=2\times7), (2744=2^3\times7^3). Step 3: 14 is composite, so write prime bases in the final form.