Prime Factorisation

Step 1: Write (35280=16\times2205). Step 2: (2205=3^2\times5\times7^2), so (35280=2^4\times3^2\times5\times7^2). Step 3: Give 2205 its complete prime form.

Step 1: (27720=8\times3465). Step 2: (8=2^3) and (3465=3^2\times5\times7\times11), so the power of 2 is 3. Step 3: Comparing gives (a=3).

Step 1: Write (31104=128\times243). Step 2: (128=2^7) and (243=3^5), so (31104=2^7\times3^5). Step 3: Comparing with the given form gives (b=5).

Step 1: (19600=140^2). Step 2: (140=2^2\times5\times7), so (19600=2^4\times5^2\times7^2). Step 3: Comparing gives (m=2).

Step 1: Write (20790=54\times385). Step 2: (54=2\times3^3) and (385=5\times7\times11), so the power of 3 is 3. Step 3: Comparing gives (p=3).

Step 1: Calculate (2^3=8) and (3^2=9). Step 2: (8\times9\times7\times11=5544). Step 3: Simplify powers first and then multiply.

Step 1: Calculate (2^5=32) and (3^3=27). Step 2: (32\times27\times7=6048). Step 3: Find the values of higher powers separately.

Step 1: Calculate (3^2=9), (5^2=25), and (7^2=49). Step 2: (9\times25\times49=11025). Step 3: In this square form, all exponents are 2, so multiply carefully.

Step 1: Calculate (2^4=16) and (7^2=49). Step 2: (16\times3\times5\times49=11760). Step 3: Calculating powers first makes the work easier.

Step 1: Calculate (2^7=128) and (3^5=243). Step 2: (128\times243=31104). Step 3: Simplifying both powers first is the correct method.

Step 1: For a perfect square, all exponents must be even. Step 2: The powers of 2 and 7 are odd. Step 3: Multiplying by (2\times7=14) makes all exponents even.

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

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: Powers 4, 2, and 1 must become 6, 3, and 3 respectively. Step 3: The multiplier is (2^2\times3\times7^2=588).

Step 1: For a perfect cube, exponents should be multiples of 3. Step 2: Reduce 8 to 6, 5 to 3, 4 to 3, and 2 to 0 for the smallest divisor. Step 3: So the divisor is (2^2\times3^2\times5\times11^2).

Step 1: For divisibility, every prime power of the divisor must be available in the number. Step 2: (n) has power 4 of 3, but (3^5) needs power 5. Step 3: Therefore, (n) is not divisible by (3^5).

Step 1: For divisibility, each exponent in the divisor must be less than or equal to the corresponding exponent in the number. Step 2: (2^7), (3^2), and (5^4) are all available in (n). Step 3: Therefore, (n) must be divisible by (2^7\times3^2\times5^4).

Step 1: To count with repetition, add the exponents. Step 2: (4+2+2+2=10). Step 3: Keep the difference between distinct prime count and repeated prime count clear.

Step 1: While counting distinct prime factors, do not add exponents. Step 2: The prime bases are 2, 3, 11, and 13. Step 3: Therefore, the number of distinct prime factors is 4.

Step 1: In multiplication, powers of the same prime base are added. Step 2: The power of 3 in (a) is 2 and in (b) is 4. Step 3: In (ab), the power of 3 will be (2+4=6).

Step 1: Powers with the same base 5 are added in multiplication. Step 2: The power of 5 in (x) is 3 and in (y) is 5. Step 3: The total power will be (3+5=8).