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**Hint:**First of all find the prime factorization of $108$ and$192$ individually.(example: prime factorization of $48$ is $2 \times 2 \times 2 \times 2 \times 3$. Add the powers of prime numbers with the same base(example: $48 = {2^4} \times {3^1}$). Then multiply the prime factorization of both given numbers.

**Complete step-by-step answer:**Step-1

Prime factorization of $108$ is $2 \times 2 \times 3 \times 3 \times 3$

${a^b} \times {a^c} = {a^{b + c}}$

${a^1} \times {a^1} \times ... \times {a^1} = {a^{1 + 1 + ... + 1}} = {a^n}$ when $a$ is multiplied by itself $n$ times

Use the above formula for adding powers of numbers having same base

On adding powers of primes having same base we can rewrite prime factorization of $108$ as ${2^2} \times {3^3}$

Step-2

Prime factorization of $192$ is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3$

Again use the formula for adding powers given in step-1

We can rewrite prime factorization of $192$ as ${2^6} \times {3^1}$

Step-3

From step-1 and step-2 write the prime factorization of $108 \times 192$

$ \Rightarrow 108 \times 192 = {2^2} \times {3^3} \times {2^6} \times {3^1}$

$ \Rightarrow 108 \times 192 = {2^2} \times {2^6} \times {3^3} \times {3^1}$

${a^b} \times {a^c} = {a^{b + c}}$

Use above formula for adding powers of numbers having same base

$ \Rightarrow 108 \times 192 = {2^{2 + 6}} \times {3^{3 + 1}}$

$ = {2^8} \times {3^4}$

Step-4

Sum of the powers of the prime factors of $182 \times 192$ is

(powers of prime number $2$) + (powers of prime number $3$ )

$ = 8 + 4$

$ = 12$

So the sum of the powers of the primes of $108 \times 192$ is $12$ .

**Hence option (D) is the correct answer.**

**Note:**Don’t multiply the given numbers. find the prime factorization of the numbers individually.

Verify the prime factorization of a number by multiplying the prime factors so that it is equal to the required number.

Or

From step-1 add the powers of prime factors of $108$

As $108 = {2^2} \times {3^3}$

$ \Rightarrow $ sum of powers of prime factors of $108$ is $2 + 3 = 5$

Now, from step-2 add the powers of prime factors of $192$

As $192 = {2^6} \times {3^1}$

$ \Rightarrow $ sum of the powers of prime factors of $192$ is $6 + 1 = 7$

Now, the sum of powers of prime factors of $108 \times 192 = $

(sum of powers of prime factors of $108$) $ + $ (sum of powers of prime factors of $192$)

$ = 5 + 7$

$ = 12$.

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