The equation \( 7(7^5 + 7^3) = 7^6 + 7^4 \) demonstrates the **distributive property** of exponents in algebra.

Here’s how it works:

### Step-by-Step Explanation:

1. **Distribute the Base**:
   - The left-hand side of the equation is \( 7 \times (7^5 + 7^3) \).

2. **Apply the Distributive Property**:
   - Distribute \( 7 \) across the terms inside the parentheses:
   \[
   7 \times 7^5 + 7 \times 7^3 = 7^6 + 7^4
   \]

3. **Use the Property of Exponents**:
   - When you multiply powers with the same base, you add the exponents. Therefore:
   \[
   7 \times 7^5 = 7^{1+5} = 7^6
   \]
   \[
   7 \times 7^3 = 7^{1+3} = 7^4
   \]

4. **Simplified Result**:
   - This gives us:
   \[
   7^6 + 7^4
   \]

This confirms that the original equation \( 7(7^5 + 7^3) = 7^6 + 7^4 \) correctly applies the distributive property combined with the property of exponents.

### Conclusion:
The equation illustrates the **distributive property** in the context of exponentiation, along with the rule for multiplying powers of the same base.

Would you like more details on this property or have other questions?

Here are 5 related questions:
1. What is the power of a product property in exponents?
2. How can you simplify expressions using the properties of exponents?
3. What happens when you raise a power to another power?
4. Can you provide an example where the distributive property does not apply?
5. How do logarithms relate to the properties of exponents?

**Tip**: Always look for opportunities to apply the distributive property, as it can simplify expressions and equations significantly.

Explore how the distributive property applies in exponentiation with a detailed explanation of the equation 7(7^5 + 7^3) = 7^6 + 7^4. Learn about exponentiation rules and the application of the distributive property. Suitable for Grades 7-9.

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