Question

The number of prime factors in the expression $(6{)}^4\times (8{)}^6\times (10{)}^8\times (12{)}^{10}$ is

• A. $48$
• B. $64$
• C. $72$
• D. $80$

Answer 1

Answer: (C)

$72$

We solve the given expression as follows:

${\left(6\right)}^4\times {\left(8\right)}^6\times {\left(10\right)}^8\times {\left(12\right)}^{10}={(2\times 3)}^4\times {\left(2^3\right)}^6\times {(2\times 5)}^8\times {(2^2\times 3)}^{10}={(2^4\times 3^4)}^4\times {\left(2^{18}\right)}^6\times {(2^8\times 5^8)}^8\times {(2^{20}\times 3^{10})}^{10}=2^4\times 3^4\times 2^{18}\times 2^8\times 5^8\times 2^{20}\times 3^{10}=2^{4+18+8+20}\times 3^{4+10}\times 5^8\{\because a^m\times a^n=a^{m+n}\}=2^{50}\times 3^{14}\times 5^8$

Here, the prime factors are $2,3$ and $5$, therefore, the number of prime factors are:

$50+14+8=72$

Hence, the number of prime factors is $72$.