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The image you uploaded contains a problem set asking to simplify each expression as a single exponential expression without evaluating the result. Let's solve these problems one by one.

### Key rule to use:
- When multiplying exponents with the same base, add the exponents:  
  \( a^m \times a^n = a^{m+n} \)
- When dividing exponents with the same base, subtract the exponents:  
  \( \frac{a^m}{a^n} = a^{m-n} \)

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### Let's go through each part:

#### a. \( 10^4 \times 10^6 \)
- Apply the rule: \( 10^{4+6} = 10^{10} \)

#### b. \( 10^3 \times 10^{-9} \)
- Apply the rule: \( 10^{3+(-9)} = 10^{-6} \)

#### c. \( 10^{-7} \times 10^{-8} \)
- Apply the rule: \( 10^{-7 + (-8)} = 10^{-15} \)

#### d. \( \frac{10^8}{10^3} \)
- Apply the rule: \( 10^{8-3} = 10^5 \)

#### e. \( \frac{10^4}{10^9} \)
- Apply the rule: \( 10^{4-9} = 10^{-5} \)

#### f. \( \frac{10^{-7}}{10^4} \)
- Apply the rule: \( 10^{-7-4} = 10^{-11} \)

#### g. \( \frac{10^8}{10^{-4}} \)
- Apply the rule: \( 10^{8-(-4)} = 10^{12} \)

#### h. \( \frac{10^{-6}}{10^{-8}} \)
- Apply the rule: \( 10^{-6-(-8)} = 10^{2} \)

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### Summary of the answers:
- **a.** \( 10^{10} \)
- **b.** \( 10^{-6} \)
- **c.** \( 10^{-15} \)
- **d.** \( 10^5 \)
- **e.** \( 10^{-5} \)
- **f.** \( 10^{-11} \)
- **g.** \( 10^{12} \)
- **h.** \( 10^{2} \)

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Here are some related questions you can explore:
1. What is the rule for dividing powers with the same base?
2. How do you handle negative exponents when simplifying?
3. How do fractional exponents work in similar situations?
4. How do you apply these rules to non-base-10 numbers?
5. How can these exponent rules be used in scientific notation?

**Tip:** When multiplying or dividing numbers with exponents, always ensure the bases are the same before applying the exponent rules.

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

Learn how to simplify exponential expressions using exponent rules. This problem set covers key algebraic concepts like multiplying and dividing exponents with the same base. Suitable for students in Grades 7-9.

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