Find the total number of prime factors of the number \(5^8 \times 6^7 \times 11^4\).
26
Solution:
Factorise every base into primes:
\(5^8\) contributes eight 5s; \(6^7=(2\times3)^7=2^7\times3^7\) contributes seven 2s and seven 3s; \(11^4\) contributes four 11s.
The complete prime factorisation is:
\[ 2^7\times3^7\times5^8\times11^4 \]
The total number of prime factors (counted with multiplicity) is:
\[ 7+7+8+4=26 \]
Hence, the total number of prime factors is: 26
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