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Let's break down the given question step-by-step.

We are asked to find which expressions are equivalent to:

\[
8^3 \cdot 10^6
\]

### Option A:
\[
(8 \cdot 10^3)^3
\]
Simplifying:
\[
= 8^3 \cdot 10^{3 \times 3} = 8^3 \cdot 10^9
\]
This is **not equivalent** to \( 8^3 \cdot 10^6 \).

### Option B:
\[
\frac{8^5 \cdot 10^{15}}{10^9 \cdot 8^2}
\]
Simplifying:
\[
= \frac{8^5}{8^2} \cdot \frac{10^{15}}{10^9} = 8^{5-2} \cdot 10^{15-9} = 8^3 \cdot 10^6
\]
This **is equivalent** to \( 8^3 \cdot 10^6 \).

### Option C:
\[
\frac{(8 \cdot 10^6)}{8^2}
\]
Simplifying:
\[
= \frac{8^1 \cdot 10^6}{8^2} = 8^{1-2} \cdot 10^6 = 8^{-1} \cdot 10^6
\]
This is **not equivalent** to \( 8^3 \cdot 10^6 \).

### Option D:
\[
8 \cdot 10^4 \cdot (8 \cdot 10)^2
\]
Simplifying:
\[
= 8 \cdot 10^4 \cdot (8^2 \cdot 10^2) = 8 \cdot 10^4 \cdot 64 \cdot 10^2 = 8 \cdot 64 \cdot 10^{4+2} = 512 \cdot 10^6
\]
This is **not equivalent** to \( 8^3 \cdot 10^6 \).

### Option E:
\[
\frac{(8 \cdot 10^5)^3}{10^3}
\]
Simplifying the numerator first:
\[
(8 \cdot 10^5)^3 = 8^3 \cdot 10^{5 \times 3} = 8^3 \cdot 10^{15}
\]
Now, the whole expression becomes:
\[
\frac{8^3 \cdot 10^{15}}{10^3} = 8^3 \cdot 10^{15-3} = 8^3 \cdot 10^{12}
\]
This is **not equivalent** to \( 8^3 \cdot 10^6 \).

### Conclusion:
The correct expression equivalent to \( 8^3 \cdot 10^6 \) is **Option B**.

Would you like more details or have any questions?

---

Here are five related questions to further your understanding:
1. How do we apply the properties of exponents when simplifying expressions like this?
2. What are the key exponent rules used in these calculations?
3. Why does option B simplify correctly, but the others do not?
4. How can you quickly identify when two expressions with exponents are equivalent?
5. How do negative exponents impact the simplification of expressions?

**Tip:** Always use exponent rules carefully to simplify each part of an expression before comparing it to the target value.

Which expressions are equivalent to 8^3 * 10^6? Select all that apply.

Explore how to simplify and identify equivalent expressions to 8^3 * 10^6 using exponent rules and algebraic simplifications. Understand which expressions are mathematically equivalent to the given term with step-by-step explanations.

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