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The given expression is:

\[
\frac{2^6 \cdot 10^{18}}{2^{24} \cdot 25^9}
\]

Let's simplify it step by step.

### Step 1: Break down the powers of 10 and 25.

We know that:
\[
10^{18} = (2 \cdot 5)^{18} = 2^{18} \cdot 5^{18}
\]
and
\[
25^9 = (5^2)^9 = 5^{18}.
\]

### Step 2: Substitute these into the expression.

Now, rewrite the expression:
\[
\frac{2^6 \cdot (2^{18} \cdot 5^{18})}{2^{24} \cdot 5^{18}}.
\]

### Step 3: Simplify the powers of 2 and 5.

- For powers of 2: \(2^6 \cdot 2^{18} = 2^{24}\).
- In the denominator, we already have \(2^{24} \cdot 5^{18}\).

So the expression becomes:
\[
\frac{2^{24} \cdot 5^{18}}{2^{24} \cdot 5^{18}}.
\]

### Step 4: Cancel out identical terms.

Since both the numerator and the denominator are the same, they cancel each other out, leaving:
\[
1.
\]

Thus, the value of the expression is \(1\).

---

Would you like further details or have any questions? Here are five related questions to expand on the topic:

1. How do you simplify expressions involving powers of prime numbers?
2. What are some other techniques to simplify algebraic fractions?
3. Can this type of simplification be applied to variables instead of numbers?
4. How do we handle exponents when they involve different bases?
5. How can we visualize or understand exponent rules better?

**Tip**: When simplifying expressions with powers, look for common bases and use exponent rules to combine or cancel terms efficiently.

Find the value of the expression: \( \frac{2^6 \cdot 10^{18}}{2^{24} \cdot 25^9} \)

Learn how to simplify the expression \( \frac{2^6 \cdot 10^{18}}{2^{24} \cdot 25^9} \) using the laws of exponents. This detailed solution demonstrates breaking down powers of numbers and canceling terms. Ideal for students in Grades 9-11.

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