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Let's calculate the given expression step by step. The expression is:

\[
\frac{(-2)^8 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10}
\]

### Step 1: Simplify \((-2)^8\)
Since \((-2)^8\) means \((-2)\) raised to the power of 8, and any even power of a negative number is positive:
\[
(-2)^8 = 256
\]

### Step 2: Simplify the powers of 5
We have \(5^3\) in the numerator and \(5^4\) in the denominator. Using the property \(a^m / a^n = a^{m-n}\):
\[
\frac{5^3}{5^4} = 5^{3-4} = 5^{-1} = \frac{1}{5}
\]

### Step 3: Simplify the powers of 2
In the denominator, we have \(2^{10}\). Since 256 in the numerator can be written as \(2^8\):
\[
\frac{2^8}{2^{10}} = 2^{8-10} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\]

### Step 4: Simplify the expression
Now we substitute back with the simplified terms:
\[
\frac{256 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10} = \frac{\frac{1}{5} \cdot \frac{1}{4}}

Calculate \( \frac{(-2)^8 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10} \)

Step-by-step solution for calculating \( \frac{(-2)^8 \cdot 5^3}{5^4 \cdot 2^{10} \cdot 10} \), involving simplification of exponential expressions, exponent rules, and simplification of terms. Suitable for Grades 9-12.

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