https://file.aiquickdraw.com/mathbot/file/1728206408646lmqbskrk.jpg

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.

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

Learn how to simplify the expression \( \frac{35^6 \cdot 15^4 \cdot 21^5}{3^{14} \cdot 5^{10} \cdot 7^{16}} \) using properties of exponents and prime factorization. This solution covers exponent rules, prime factorization, and simplifying complex expressions, suitable for high school students.

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