Prime Factorisation

Step 1: In a perfect square, all exponents should be even. Step 2: The powers of 3 and 7 are odd. Step 3: Multiplying by (3\times7=21) makes both powers even.

Step 1: For a perfect square, exponents should be even. Step 2: The powers of 2 and 5 are odd. Step 3: Dividing by (2\times5) makes the powers 8 and 6.

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 5, 7, 4, and 2 must become 6, 9, 6, and 3. Step 3: The smallest multiplier is (2\times3^2\times5^2\times13).

Step 1: For a perfect cube, exponents should be multiples of 3. Step 2: Reducing 10 to 9, 8 to 6, 5 to 3, and 4 to 3 is the smallest way. Step 3: Therefore, the divisor is (2\times3^2\times5^2\times7).

Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 2 of 11, but (11^3) needs power 3. Step 3: Therefore, (n) is not divisible by (11^3).

Step 1: For divisibility, exponents in the divisor must not exceed those in the given number. Step 2: (2^9), (3^4), (7^3), and 13 are all available in (n). Step 3: Therefore, (n) must be divisible by the first option.

Step 1: To count with repetition, add the exponents. Step 2: (9+7+4+3+2=25). Step 3: Keep the number of bases and the total count with repetition separate.

Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 7, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 5.

Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 3 in (a) is 4 and in (b) is 5. Step 3: In (ab), the power of 3 will be (4+5=9).

Step 1: Powers with the same base 13 are added in multiplication. Step 2: The power of 13 in (x) is 1 and in (y) is 3. Step 3: The total power will be (1+3=4).

Step 1: Calculate (2^5=32), (3^4=81), and (7^2=49). Step 2: (32\times81\times49=127008). Step 3: Solve all three powers separately first.

Step 1: Calculate (2^3=8), (3^5=243), and (7^2=49). Step 2: (8\times243\times49=95256). Step 3: Simplify higher powers first.

Step 1: In the final form, every base must be prime. Step 2: In the first option, bases 2, 3, 5, and 7 are prime. Step 3: 8, 135, 49, 27, 245, and 216 are composite, so they are not final forms.

Step 1: In an incomplete form, composite bases remain. Step 2: 81 and 49 are composite bases. Step 3: (2^5\times81\times49) must be changed into (2^5\times3^4\times7^2).

Step 1: (2^6=64), (3^3=27), and (5^2=25). Step 2: (64\times27\times25=43200). Step 3: Solving powers first helps find the correct option quickly.

Step 1: Calculate (2^2=4), (3^2=9), (5^2=25), and (11^2=121). Step 2: (4\times9\times25\times121=108900). Step 3: This is a square form, so observe the powers carefully.

Step 1: Write (158760=8\times19845). Step 2: (19845=3^4\times5\times7^2), so (158760=2^3\times3^4\times5\times7^2). Step 3: Break 19845 completely.

Step 1: Write (217800=8\times27225). Step 2: (27225=3^2\times5^2\times11^2), so (217800=2^3\times3^2\times5^2\times11^2). Step 3: Convert 27225 into prime powers.

Step 1: (279936) can be written as (128\times2187). Step 2: (128=2^7) and (2187=3^7), so (279936=2^7\times3^7). Step 3: Do not leave 128 and 2187 in the final form.

Step 1: In final prime factorisation, every base should be prime. Step 2: (8=2^3) and (27225=3^2\times5^2\times11^2). Step 3: Therefore, the final form is (2^3\times3^2\times5^2\times11^2).