Prime Factorisation

Step 1: Calculate (2^6=64), (3^4=81), and (11^2=121). Step 2: (64\times81\times121=627264). Step 3: Solve all three powers separately first.

Step 1: Calculate (2^4=16), (3^3=27), (5^2=25), and (7^2=49). Step 2: (16\times27\times25\times49=529200). Step 3: Do the multiplication step by step.

Step 1: In the final form, bases must be prime. Step 2: In the first option, bases 2, 3, 5, and 7 are prime. Step 3: 32, 81, 25, 405, 35, and 160 are composite, so they are not final forms.

Step 1: In an incomplete form, composite bases remain. Step 2: 81 and 121 are composite bases. Step 3: (2^6\times81\times121) must be changed into (2^6\times3^4\times11^2).

Step 1: Take (2^4=16), (3^5=243), (5^2=25), and (7). Step 2: (16\times243\times25\times7=680400). Step 3: Solving powers first helps find the correct option quickly.

Step 1: Calculate (2^3=8), (3^3=27), and (5^2=25). Step 2: (8\times27\times25\times7\times11=415800). Step 3: When there are many factors, multiply in small groups.

Step 1: Write the number using prime-base groups. Step 2: (16\times27\times25\times49\times11=2^4\times3^3\times5^2\times7^2\times11). Step 3: Avoid decimal-based options and write prime bases.

Step 1: Write (254016=64\times3969). Step 2: (64=2^6) and (3969=3^4\times7^2), so (254016=2^6\times3^4\times7^2). Step 3: Convert 3969 into prime powers.

Step 1: Write (2138400=32\times66825). Step 2: (66825=3^5\times5^2\times11), so (2138400=2^5\times3^5\times5^2\times11). Step 3: Give 66825 its complete prime form.

Step 1: In final prime factorisation, every base should be prime. Step 2: (64=2^6) and (3969=3^4\times7^2). Step 3: Therefore, the final form is (2^6\times3^4\times7^2).

Step 1: In prime factorisation, add 1 to each exponent and multiply. Step 2: Here the number of factors is ((3+1)(2+1)(1+1)=24). Step 3: In exams, identify the exponents first and then multiply carefully.

Step 1: The bases in prime factorisation are the prime factors. Step 2: Among (2,3,7), the smallest prime number is (2). Step 3: For the smallest prime factor, look at the base, not the exponent.

Step 1: Evaluate the powers first. Step 2: (2^2 \times 3 \times 5^2=4 \times 3 \times 25=300). Step 3: In such questions, solve powers before multiplying.

Step 1: In HCF, take the smaller exponent of common prime factors. Step 2: The exponents of (3) are (2) and (4), so the smaller one is (2). Step 3: For HCF, always choose the minimum exponent.

Step 1: In LCM, take the larger exponent of each prime factor. Step 2: The exponents of (5) are (2) and (4), so the larger one is (4). Step 3: Remember that LCM uses maximum exponents.

Step 1: Write (360=36 \times 10). Step 2: Since (36=2^2 \times 3^2) and (10=2 \times 5), (360=2^3 \times 3^2 \times 5). Step 3: Verify by multiplying back to the original number.

Step 1: Prime factorise (72). Step 2: (72=8 \times 9=2^3 \times 3^2), so (a=3). Step 3: To find an unknown exponent, compare prime factorisations on both sides.

Step 1: Distinct prime factors are counted by their prime bases. Step 2: The bases are (2,3,5), so there are (3) distinct prime factors. Step 3: Do not count exponents as separate factors.

Step 1: (2^4 \times 3 \times 7=16 \times 21=336). Step 2: (336,672,1008,1344) are all divisible by (336), so none is actually not divisible. Step 3: In exams, read negative wording and test every option carefully.

Step 1: Evaluate the power first. Step 2: (2^3=8), so (N=8 \times 3 \times 11=264). Step 3: Group smaller products to calculate faster.