https://file.aiquickdraw.com/mathbot/file/1725827141696lsasm0t0.jpg

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:  
\[ 10^{-6 - (-5)} = 10^{-1} \]

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### 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?

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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.

Simplify each expression so it is written as only one exponential expression. Do not evaluate to a number.

Learn how to simplify expressions involving exponents by applying the rules of multiplication and division of powers. This exercise is suitable for students in Grades 7-9, helping them strengthen their understanding of exponential expressions.

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