Number of prime factors in $$\frac{6^{12} \times (35)^{28} \times (15)^{16}}{(14)^{12} \times (21)^{11}}$$ is

Concept & Formula To find the total number of prime factors in a complex fraction, we must express every composite base as a product of prime numbers. The fundamental laws of indices applied here are: $$(a \times b)^m = a^m \times b^m$$ $$\frac{a^m}{a^n} = a^{m-n}$$ Once simplified to the form $P_1^a \times P_2^b \times P_3^c$, the total number of prime factors is the sum of the positive exponents: $a + b + c$. Step-by-step Solution First, let's break down the numerator into prime bases: $6^{12} = (2 \times 3)^{12} = 2^{12} \times 3^{12}$ $35^{28} = (5 \times 7)^{28} = 5^{28} \times 7^{28}$ $15^{16} = (3 \times 5)^{16} = 3^{16} \times 5^{16}$ Combining the numerator bases: Base 2: $12$ Base 3: $12 + 16 = 28$ Base 5: $28 + 16 = 44$ Base 7: $28$ Numerator = $2^{12} \times 3^{28} \times 5^{44} \times 7^{28}$ Next, let's break down the denominator into prime bases: $14^{12} = (2 \times 7)^{12} = 2^{12} \times 7^{12}$ $21^{11} = (3 \times 7)^{11} = 3^{11} \times 7^{11}$ Combining the denominator bases: Base 2: $12$ Base 3: $11$ Base 7: $12 + 11 = 23$ Denominator = $2^{12} \times 3^{11} \times 7^{23}$ Now, divide the numerator by the denominator by subtracting the exponents of like bases: Power of 2: $12 - 12 = 0$ Power of 3: $28 - 11 = 17$ Power of 5: $44 - 0 = 44$ Power of 7: $28 - 23 = 5$ The fully simplified expression is: $2^0 \times 3^{17} \times 5^{44} \times 7^5$ Total number of prime factors = $17 + 44 + 5 = 66$. Exam Strategy & Shortcut Avoid writing out the intermediate steps. Set up a quick tally chart for prime numbers 2, 3, 5, and 7. For each prime, mentally scan the expression. For 7: It appears in 35 (power 28, numerator), 14 (power 12, denominator), and 21 (power 11, denominator). Net power = $28 - 12 - 11 = 5$. Do this rapidly for all primes and sum them up. Common Pitfall A frequent mistake is missing a hidden prime factor when decomposing numbers, or forgetting to distribute the outer exponent to BOTH inner prime bases (e.g., writing $15^{16}$ as $3 \times 5^{16}$ instead of $3^{16} \times 5^{16}$). Always use brackets when splitting composite numbers! Final Answer **Therefore, the correct answer is 66.**