Let's simplify the given expression step-by-step:
[tex]\[
\frac{2x^3 + 8x^3}{(5x)(2x)^2}
\][/tex]
### Step 1: Simplify the Numerator
First, we combine the like terms in the numerator:
[tex]\[
2x^3 + 8x^3 = (2 + 8)x^3 = 10x^3
\][/tex]
So, the expression becomes:
[tex]\[
\frac{10x^3}{(5x)(2x)^2}
\][/tex]
### Step 2: Simplify the Denominator
Next, we simplify the denominator. Start with simplifying [tex]\((2x)^2\)[/tex]:
[tex]\[
(2x)^2 = 2^2 \cdot x^2 = 4x^2
\][/tex]
Now, multiply this result by [tex]\(5x\)[/tex]:
[tex]\[
(5x)(4x^2) = 5x \cdot 4x^2 = (5 \cdot 4) \cdot (x \cdot x^2) = 20x^3
\][/tex]
### Step 3: Simplify the Fraction
Now we have the expression:
[tex]\[
\frac{10x^3}{20x^3}
\][/tex]
Since [tex]\(10x^3\)[/tex] and [tex]\(20x^3\)[/tex] both have a common factor of [tex]\(10x^3\)[/tex], we can simplify it:
[tex]\[
\frac{10x^3}{20x^3} = \frac{10}{20} \cdot \frac{x^3}{x^3} = \frac{1}{2} \cdot 1 = \frac{1}{2}
\][/tex]
### Final Answer
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
