The image asks to simplify each expression into a single exponential expression without evaluating the numbers. Let's go through the problems one by one.

Rules to Remember:

• Multiplication of powers: $a^m \times a^n = a^{m+n}$
• Division of powers: $\frac{a^m}{a^n} = a^{m-n}$
• Negative exponents: $a^{-n} = \frac{1}{a^n}$

Now let's simplify:

a. $10^4 \times 10^6$

Using the multiplication rule:
$10^{4 + 6} = 10^{10}$

b. $10^3 \times 10^{-9}$

Using the multiplication rule:
$10^{3 + (-9)} = 10^{-6}$

c. $10^7 \times 10^{-8}$

Using the multiplication rule:
$10^{7 + (-8)} = 10^{-1}$

d. $\frac{10^8}{10^4}$

Using the division rule:
$10^{8 - 4} = 10^4$

e. $\frac{10^4}{10^9}$

Using the division rule:
$10^{4 - 9} = 10^{-5}$

f. $\frac{10^{-7}}{10^4}$

Using the division rule:
$10^{-7 - 4} = 10^{-11}$

g. $\frac{10^8}{10^{-3}}$

Using the division rule:
$10^{8 - (-3)} = 10^{11}$

h. $\frac{10^{-6}}{10^{-5}}$

Using the division rule:
$10^{-6 - (-5)} = 10^{-1}$

Final simplified expressions:

a. $10^{10}$
b. $10^{-6}$
c. $10^{-1}$
d. $10^4$
e. $10^{-5}$
f. $10^{-11}$
g. $10^{11}$
h. $10^{-1}$

Would you like further explanations or any clarifications?

Here are 5 related questions to expand your understanding:

1. What happens if the bases are different when multiplying exponential terms?
2. How would the simplifications change if there were coefficients in front of the exponents (e.g., $3 \times 10^4 \times 5 \times 10^6$)?
3. Can you explain the process of simplifying $\frac{a^m}{a^{-n}}$?
4. How would you simplify expressions with multiple operations, such as $10^3 \times 10^{-2} \div 10^4$?
5. What happens when an exponent has a fractional value, such as $10^{3/2}$?

Tip:

Always remember that the exponent rules only apply when the base (in this case, 10) is the same.