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To determine which of the given expressions has a value of 1, we can simplify each of them.

### 1. \( 7^{-6} \cdot 7^5 \)
Using the property of exponents, \( a^m \cdot a^n = a^{m+n} \):
\[
7^{-6} \cdot 7^5 = 7^{-6+5} = 7^{-1} = \frac{1}{7} \neq 1
\]
This is not equal to 1.

### 2. \( \frac{10^3}{10^2} \)
Using the property of exponents, \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{10^3}{10^2} = 10^{3-2} = 10^1 = 10 \neq 1
\]
This is not equal to 1.

### 3. \( \frac{(5^4)^2}{5^8} \)
Simplify the numerator using the power of a power rule, \( (a^m)^n = a^{mn} \):
\[
(5^4)^2 = 5^{4 \times 2} = 5^8
\]
Thus, the expression becomes:
\[
\frac{5^8}{5^8} = 1
\]
This is equal to 1.

### 4. \( \frac{3^2 \cdot 3^5}{3^{10}} \)
Using the properties of exponents:
\[
3^2 \cdot 3^5 = 3^{2+5} = 3^7
\]
So the expression becomes:
\[
\frac{3^7}{3^{10}} = 3^{7-10} = 3^{-3} = \frac{1}{3^3} \neq 1
\]
This is not equal to 1.

### Conclusion:
The third expression, \( \frac{(5^4)^2}{5^8} \), has a value of 1.

Would you like more details on any of the steps?

### Related Questions:
1. How do exponent rules apply to multiplication and division of powers?
2. Can negative exponents ever result in values of 1?
3. What is the general property for simplifying powers raised to another power?
4. How does the concept of zero exponent work with base values?
5. Why do certain combinations of exponents result in 1?

### Tip:
Always use exponent properties like \( a^m \cdot a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \) to simplify expressions efficiently.

Learn how to identify which expressions have a value of 1 by applying the rules of exponents. Explore the simplification of powers and exponents through examples like (5^4)^2 / 5^8. Suitable for Grades 8-10.

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