The expression in the image is:

$\frac{35^6 \cdot 15^4 \cdot (21^5)}{3^{14} \cdot 5^{10} \cdot 7^{16}}.$

To simplify, we will first use the properties of exponents to break this down:

1. Apply the exponent rules where appropriate: $a^m \cdot a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$.

Let's rewrite the expression:

$\frac{35^6 \cdot 15^4 \cdot 21^5}{3^{14} \cdot 5^{10} \cdot 7^{16}}.$

Step 1: Prime Factorization of the Numbers

• $35 = 5 \cdot 7$, so $35^6 = 5^6 \cdot 7^6$.
• $15 = 3 \cdot 5$, so $15^4 = 3^4 \cdot 5^4$.
• $21 = 3 \cdot 7$, so $21^5 = 3^5 \cdot 7^5$.

Step 2: Substitute these into the expression

$\frac{(5^6 \cdot 7^6) \cdot (3^4 \cdot 5^4) \cdot (3^5 \cdot 7^5)}{3^{14} \cdot 5^{10} \cdot 7^{16}}.$

Step 3: Combine like terms (using the exponent rules)

• For $3$: $3^{4+5} = 3^9$.
• For $5$: $5^{6+4} = 5^{10}$.
• For $7$: $7^{6+5} = 7^{11}$.

Now the expression looks like this:

$\frac{3^9 \cdot 5^{10} \cdot 7^{11}}{3^{14} \cdot 5^{10} \cdot 7^{16}}.$

Step 4: Simplify each term by subtracting exponents

• For $3$: $\frac{3^9}{3^{14}} = 3^{-5} = \frac{1}{3^5}$.
• For $5$: $\frac{5^{10}}{5^{10}} = 1$.
• For $7$: $\frac{7^{11}}{7^{16}} = 7^{-5} = \frac{1}{7^5}$.

Thus, the simplified expression becomes:

$\frac{1}{3^5 \cdot 7^5}.$

Final Step: Combine the terms

$\frac{1}{(3 \cdot 7)^5} = \frac{1}{21^5}.$

So, the final simplified expression is:

$\boxed{\frac{1}{21^5}}.$

Would you like more details on any of these steps? Here are a few related questions to deepen your understanding:

1. How can prime factorization help in simplifying exponent problems?
2. What are the properties of exponents and how do they apply in division?
3. Can you explain why $3^9 / 3^{14} = 3^{-5}$?
4. How would the problem change if any base had a negative exponent?
5. Why does $5^{10} / 5^{10}$ equal 1, and is this true for all bases?

Tip: Always break down numbers into their prime factors when dealing with exponents to make simplifications easier.