Sure, let's analyze the given expression step-by-step to simplify and find the equivalent expression.
Given expression:
[tex]\[ 4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8} \][/tex]
First, let's rewrite the square root terms for better simplification:
[tex]\[ \sqrt{5 x^4} \text{ and } \sqrt{5 x^8} \][/tex]
Since [tex]\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)[/tex], we can rewrite them as:
[tex]\[ \sqrt{5 x^4} = \sqrt{5} \cdot \sqrt{x^4} = \sqrt{5} \cdot x^2 \][/tex]
[tex]\[ \sqrt{5 x^8} = \sqrt{5} \cdot \sqrt{x^8} = \sqrt{5} \cdot x^4 \][/tex]
Now, substitute these back into the original expression:
[tex]\[ 4 x^2 \cdot (\sqrt{5} \cdot x^2) \cdot 3 \cdot (\sqrt{5} \cdot x^4) \][/tex]
Combine the constants and the square root terms:
[tex]\[ 4 \cdot 3 \cdot x^2 \cdot x^2 \cdot x^4 \cdot \sqrt{5} \cdot \sqrt{5} \][/tex]
Since [tex]\( \sqrt{5} \cdot \sqrt{5} = 5 \)[/tex]:
[tex]\[ 4 \cdot 3 \cdot 5 \cdot x^2 \cdot x^2 \cdot x^4 \][/tex]
Simplify the constants:
[tex]\[ 4 \cdot 3 \cdot 5 = 60 \][/tex]
Combine the powers of [tex]\( x \)[/tex]:
[tex]\[ x^2 \cdot x^2 \cdot x^4 = x^{2+2+4} = x^8 \][/tex]
Thus, the expression simplifies to:
[tex]\[ 60 x^8 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{60 x^8} \][/tex]
