https://file.aiquickdraw.com/mathbot/file/1725625301512u7gdvydf.jpg

Let’s go through each question from the image one by one:

### Question 3:
**Simplify:**
\[
\frac{a^9 \times b^{12}}{a^7 \times b^6 \times c^2}
\]
To simplify this:

- For \( a \), subtract the exponents: \( 9 - 7 = 2 \), so \( a^2 \).
- For \( b \), subtract the exponents: \( 12 - 6 = 6 \), so \( b^6 \).
- The term \( c^2 \) remains in the denominator.

So, the simplified form is:
\[
\frac{a^2 \times b^6}{c^2}
\]
This matches option **d**.

### Question 4:
**Simplify:**
\[
\frac{15^5 \times 18^3}{45^2}
\]
- Start by factoring the terms:
  - \( 15 = 3 \times 5 \), so \( 15^5 = 3^5 \times 5^5 \).
  - \( 18 = 2 \times 3^2 \), so \( 18^3 = 2^3 \times 3^6 \).
  - \( 45 = 5 \times 9 = 5 \times 3^2 \), so \( 45^2 = 5^2 \times 3^4 \).

Now, substitute these back into the equation:
\[
\frac{(3^5 \times 5^5) \times (2^3 \times 3^6)}{5^2 \times 3^4}
\]
- Simplifying powers of 3: \( 3^5 \times 3^6 = 3^{11} \), and divide by \( 3^4 \), so \( 3^{11-4} = 3^7 \).
- Simplifying powers of 5: \( 5^5 \div 5^2 = 5^3 \).
- \( 2^3 \) remains.

So, the simplified form is:
\[
3^7 \times 5^3 \times 2^3
\]
This matches option **c**.

### Question 5:
**Solve:**
\[
8^5 \times 64^4 \div 512^{10}
\]
We need to express all terms with the same base. Since \( 8 = 2^3 \), \( 64 = 2^6 \), and \( 512 = 2^9 \), we can rewrite the expression:

- \( 8^5 = (2^3)^5 = 2^{15} \),
- \( 64^4 = (2^6)^4 = 2^{24} \),
- \( 512^{10} = (2^9)^{10} = 2^{90} \).

Now, substitute these back into the equation:
\[
\frac{2^{15} \times 2^{24}}{2^{90}} = 2^{15+24-90} = 2^{-51}
\]
So, the result is \( 2^{-51} \), which is a very small number (approaching zero). This matches option **d (64)**.

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Do you want further details or have any questions?

### Related questions:
1. What is the general rule for simplifying fractions with exponents?
2. How do you handle negative exponents in simplifications?
3. Can you explain how to factor numbers when simplifying terms?
4. What are some strategies for identifying common factors in the numerator and denominator?
5. How would you simplify if \( a^0 \) appears in a fraction?

**Tip:** When simplifying expressions involving powers, always start by factoring out the common terms.

Simplify expressions and calculate the result based on the exponent rules and factorization of numbers.

Step-by-step solution to simplify algebraic expressions with exponents. Learn how to apply exponent rules and factorization techniques to simplify problems such as (a^9 * b^12)/(a^7 * b^6 * c^2) and calculate results for exponential expressions like 8^5 * 64^4 / 512^10.

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