Let's simplify the given expression step by step. The original expression is:

$\frac{6^4 \times 15^4 \times 8^4}{2^{16} \times 3^8 \times 5^4}$

Step 1: Prime Factorization of Base Numbers

1. $6 = 2 \times 3$, so $6^4 = 2^4 \times 3^4$.
2. $15 = 3 \times 5$, so $15^4 = 3^4 \times 5^4$.
3. $8 = 2^3$, so $8^4 = 2^{12}$.

Substituting these into the expression:

$\frac{(2^4 \times 3^4) \times (3^4 \times 5^4) \times 2^{12}}{2^{16} \times 3^8 \times 5^4}$

Step 2: Combine Like Terms

In the numerator:

• Powers of $2$: $2^4 \times 2^{12} = 2^{16}$.
• Powers of $3$: $3^4 \times 3^4 = 3^8$.
• Powers of $5$: $5^4$.

The numerator becomes:

$2^{16} \times 3^8 \times 5^4$

The denominator is already:

$2^{16} \times 3^8 \times 5^4$

Step 3: Simplify

Cancel out all terms in the numerator and denominator:

$\frac{2^{16} \times 3^8 \times 5^4}{2^{16} \times 3^8 \times 5^4} = 1$

Final Answer:

$1$

Let me know if you'd like any additional details or examples of similar problems!

Questions for Further Practice:

1. What happens if the powers in the numerator and denominator do not completely cancel out?
2. How do you approach problems involving variables instead of numbers?
3. How can you simplify expressions with fractional exponents?
4. Can you rewrite $(a^b)^c$ into a simpler exponential form?
5. How would the solution change if $15^4$ was replaced with $10^4$?

Tip:

Always start by factoring the base numbers into primes to simplify exponential expressions more effectively.