Prime Factorisation

Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times7=504). Step 3: Evaluate prime powers first, then multiply.

Step 1: Calculate (2^4=16). Step 2: (16\times3\times11=528). Step 3: While multiplying, complete smaller products first.

Step 1: Calculate (2^2=4) and (7^2=49). Step 2: (4\times5\times49=980). Step 3: Finding powers first reduces mistakes.

Step 1: Calculate (2^6=64) and (3^3=27). Step 2: (64\times27=1728). Step 3: Simplify higher powers separately first.

Step 1: Calculate (2^2=4) and (3^2=9). Step 2: (4\times9\times5\times11=1980). Step 3: Simplifying powers first gives the answer quickly.

Step 1: (2^5=32). Step 2: (32\times3\times7=672). Step 3: Multiply all factors to convert prime factorisation into the number.

Step 1: (3^3=27). Step 2: (27\times5\times7=945). Step 3: It is better to simplify the power part first.

Step 1: In prime factorisation, bases must be prime. Step 2: In the first option, bases 2, 3, 5, and 7 are prime. Step 3: 8, 9, 35, and 4 are composite, so the other forms are not final.

Step 1: Final prime factorisation must not contain a composite factor. Step 2: 21 is composite, so (2^3\times21\times5) is not final form. Step 3: Change 21 into (3\times7).

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 (2^{11}) 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 write a power of 3 in final form.

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

Step 1: (2401) can be written as (49\times49). Step 2: Since (49=7^2), (2401=7^4). Step 3: Since 49 is composite, 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: Count the total power of 5 as 4.

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

Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5=720). Step 3: Solving powers first helps find the option quickly.

Step 1: Calculate (2^3=8), (3^2=9), and (5^2=25). Step 2: (8\times9\times25=1800). Step 3: Find each power separately and then multiply.

Step 1: (2^4=16) and (3^4=81). Step 2: (16\times81=1296). Step 3: In such calculations, simplifying powers first is safer.

Step 1: Calculate (2^2=4) and (5^4=625). Step 2: (4\times625=2500). Step 3: Finding the higher power first makes the answer easier.

Step 1: Write (2700=27\times100). Step 2: (27=3^3) and (100=2^2\times5^2), so (2700=2^2\times3^3\times5^2). Step 3: Convert both 27 and 100 into prime powers.

Step 1: Write (3080=8\times385). Step 2: (8=2^3) and (385=5\times7\times11), so (3080=2^3\times5\times7\times11). Step 3: Break 385 into prime form too.

Step 1: Write (3960=36\times110). Step 2: (36=2^2\times3^2) and (110=2\times5\times11), so (3960=2^3\times3^2\times5\times11). Step 3: Count the total power of 2 carefully.

Step 1: Calculate (2^3=8). Step 2: (8\times5\times7\times11=3080). Step 3: Whatever the order, the product remains the same.

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: In final prime factorisation, the base must be prime. Step 2: 12 is composite and (12=2^2\times3). Step 3: Therefore, (12^2) must be changed further into (2^4\times3^2).