Let's solve each of the problems shown in the image step by step.

Problem c

$\frac{5^9}{5^3 \times 5^7}$

Using the properties of exponents, we can simplify this expression. First, recall that $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$.

$5^3 \times 5^7 = 5^{3+7} = 5^{10}$ $\frac{5^9}{5^{10}} = 5^{9-10} = 5^{-1} = \frac{1}{5}$

So, the answer is: $\frac{1}{5}$

Problem d

$(6^5 \div 6^2)^7$

First, simplify inside the parentheses using the property $\frac{a^m}{a^n} = a^{m-n}$.

$6^5 \div 6^2 = 6^{5-2} = 6^3$

Then raise the result to the 7th power.

$(6^3)^7 = 6^{3 \times 7} = 6^{21}$

So, the answer is: $6^{21}$

Problem e

$6^9 \times 6^6 \div 6^2$

Using the properties of exponents, we can combine and simplify the terms:

$6^9 \times 6^6 = 6^{9+6} = 6^{15}$ $6^{15} \div 6^2 = 6^{15-2} = 6^{13}$

So, the answer is: $6^{13}$

Would you like further details or have any questions about these solutions?

Here are some related questions to expand your understanding:

1. How do the properties of exponents simplify complex expressions?
2. Can you explain the process of dividing exponents with the same base?
3. How would you simplify an expression with different bases, such as $\frac{3^4}{9^2}$?
4. What are some common mistakes to avoid when working with exponents?
5. How do you apply the power rule in exponentiation, and why does it work?
6. Can you simplify $\left(2^3 \times 4^2\right)^2$?
7. What happens when you raise a power to another power, such as $(a^m)^n$?
8. How would you handle exponents with negative bases, like $(-3)^4$?

Tip

When dealing with exponents, always pay attention to the base. Exponent rules only apply when the bases are the same.