Let's solve the expression step-by-step:

Given expression:

\[
\frac{625^3}{8} \times 5^{1/2} \div 25
\]

1. **Simplify each part of the expression:**

   - \(625^3\): Since \(625 = 5^4\), we have:

   \[
   625^3 = (5^4)^3 = 5^{12}
   \]

   - \(5^{1/2}\) is already simplified.
   - \(25 = 5^2\)

2. **Rewrite the expression in terms of powers of 5:**

\[
\frac{5^{12}}{8} \times 5^{1/2} \div 5^2
\]

3. **Combine the terms involving powers of 5:**

   - The expression becomes:

   \[
   \frac{5^{12} \times 5^{1/2}}{8 \times 5^2}
   \]

   - Combine the powers of 5 in the numerator and denominator:

   \[
   \frac{5^{12 + 1/2}}{8 \times 5^2} = \frac{5^{12.5}}{8 \times 5^2}
   \]

   - Subtract the exponent in the denominator from the exponent in the numerator:

   \[
   5^{12.5 - 2} = 5^{10.5}
   \]

   - Thus, the expression simplifies to:

   \[
   \frac{5^{10.5}}{8}
   \]

4. **Convert the exponent to a more understandable form:**

\[
5^{10.5} = 5^{10 + 0.5} = 5^{10} \times 5^{0.5}
\]

Since \(5^{0.5} = \sqrt{5}\), the expression can be written as:

\[
\frac{5^{10} \cdot \sqrt{5}}{8}
\]

5. **Final Simplified Form:**

\[
\frac{5^{10} \cdot \sqrt{5}}{8}
\]

Therefore, the simplified expression is:

\[
\frac{5^{10} \sqrt{5}}{8}
\]

Would you like more details on any step or have any questions?

### Related Questions:
1. How do you handle powers with fractional exponents?
2. What is the general rule for multiplying and dividing exponents with the same base?
3. How can we simplify expressions with mixed operations (multiplication, division, powers)?
4. How does the concept of square roots relate to fractional exponents?
5. How would the solution change if there was a different base, like 2 instead of 5?

**Tip:** When dealing with exponents, remember that multiplying powers with the same base means adding the exponents, while dividing them means subtracting the exponents.

Discover how to simplify the expression 625^3/8 × 5^1/2 ÷ 25 using properties of exponents. This detailed solution explains step-by-step methods to handle fractional exponents and simplify complex expressions, ideal for high school students in Grades 9-12.

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