Prime Factorisation

Step 1: In a perfect square, all exponents are even. Step 2: Powers of 2, 5, and 11 are odd. Step 3: Multiplying by (2\times5\times11=110) makes all exponents even.

Step 1: In a perfect square, all exponents should be even. Step 2: Only the power of 3 is odd. Step 3: Dividing by 3 makes the power of 3 equal to 4 and the number becomes a perfect square.

Step 1: In a perfect cube, exponents are multiples of 3. Step 2: Powers 5, 4, 2, and 1 must become 6, 6, 3, and 3. Step 3: The smallest multiplier is (2\times3^2\times5\times7^2).

Step 1: For a perfect cube, exponents should be multiples of 3. Step 2: Reducing 8 to 6, 7 to 6, and 4 to 3 is the smallest way. Step 3: Therefore, the divisor is (2^2\times3\times5).

Step 1: Every prime power of a divisor must be available in the number. Step 2: (n) has power 5 of 2, but (2^6) needs 6. Step 3: Therefore, (n) is not divisible by (2^6).

Step 1: For divisibility, exponents in the divisor must not exceed those in the number. Step 2: (2^5), (3), (5^4), and 11 are all available in (n). Step 3: Therefore, (n) must be divisible by this number.

Step 1: For counting with repetition, add the exponents. Step 2: (6+4+3+2=15). Step 3: Remember the difference between the number of distinct bases and the count with repetition.

Step 1: While counting distinct primes, only bases are counted. Step 2: The bases are 2, 3, 11, and 13. Step 3: Therefore, there are 4 distinct prime factors.

Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 7 in (a) is 1 and in (b) is 2. Step 3: In (ab), the power of 7 will be (1+2=3).

Step 1: Powers of the same base 11 are added in multiplication. Step 2: The power of 11 in (x) is 2 and in (y) is 1. Step 3: The total power will be (2+1=3).

Step 1: Calculate (2^4=16), (3^2=9), and (7^2=49). Step 2: (16\times9\times5\times49=35280). Step 3: Solve powers separately first.

Step 1: Calculate (2^8=256) and (3^4=81). Step 2: (256\times81=20736). Step 3: Simplify higher powers separately and multiply.

Step 1: In the final form, bases must be prime. Step 2: In the first form, the bases 2, 3, 5, 7, and 11 are prime. Step 3: 8, 9, 385, and 72 are composite, so they are not final forms.

Step 1: In an incomplete form, composite factors remain. Step 2: Both 45 and 49 are composite. Step 3: (2^4\times45\times49) must be changed into (2^4\times3^2\times5\times7^2).

Step 1: Calculate (2^4=16) and (3^2=9). Step 2: (16\times9\times5\times7=5040). Step 3: Solving powers first helps find the correct option quickly.

Step 1: Calculate (2^3=8), (5^2=25), and (7^2=49). Step 2: (8\times25\times49=9800). Step 3: Find all three powers separately.

Step 1: Write (29400=600\times49). Step 2: (600=2^3\times3\times5^2) and (49=7^2), so (29400=2^3\times3\times5^2\times7^2). Step 3: Break 600 and 49 completely.

Step 1: Write (33075=675\times49). Step 2: (675=3^3\times5^2) and (49=7^2), so (33075=3^3\times5^2\times7^2). Step 3: Write 675 and 49 as prime powers.

Step 1: (46656) can be written as (64\times729). Step 2: (64=2^6) and (729=3^6), so (46656=2^6\times3^6). Step 3: Do not leave 64 and 729 in the final form.

Step 1: In final prime factorisation, every base should be prime. Step 2: (600=2^3\times3\times5^2) and (49=7^2). Step 3: Therefore, the final form is (2^3\times3\times5^2\times7^2).