Fundamental Theorem of Arithmetic

Step 1: The product is (221\times9699690). Step 2: (221=13\times17), and 9699690 has 17 to power 1. Step 3: Therefore, the power of 17 in the product is (1+1=2).

Step 1: Write (320000=32\times10000). Step 2: (32=2^5) and (10000=10^4=2^4\times5^4). Step 3: Therefore, (320000=2^9\times5^4).

Step 1: In a perfect cube, every exponent must be a multiple of 3. Step 2: (2^{12}) is suitable, while (3^{11}) and (5^8) must be reduced to (3^9) and (5^6). Step 3: So the smallest divisor is (3^2\times5^2).

Step 1: Calculate (2^8=256) and (3^4=81). Step 2: (256\times81\times5\times7=725760). Step 3: To get the number from prime factorisation, multiply all factors.

Step 1: For two numbers, HCF (\times) LCM equals the product of the two numbers. Step 2: In prime powers, the smaller and higher exponents together give the total exponent. Step 3: Therefore, the answer is (ab).

Step 1: HCF is (2^8\times3^5). Step 2: LCM is (2^{11}\times3^8). Step 3: The ratio is (2^3\times3^3=8\times27=216).

Step 1: In (xy), exponents of the same bases are added. Step 2: Powers become (2^{16}), (3^{13}), and (5^{13}). Step 3: Counting with repetition gives (16+13+13=42).

Step 1: In LCM, take the higher power. Step 2: The powers of 5 are 5 and 8. Step 3: The higher power is 8, so the answer is 8.

Step 1: In HCF, take the smaller power. Step 2: The powers of 3 are 6 and 9. Step 3: The smaller power is 6, so the answer is 6.