To simplify the expression \((2x^2y^3)^4 \div (2yx^3) \cdot (2xy^3)\), we can break it down step by step.
Step 1: Apply the power of a product rule
First, simplify \((2x^2y^3)^4\): \[ (2x^2y^3)^4 = 2^4 \cdot (x^2)^4 \cdot (y^3)^4 = 16 \cdot x^8 \cdot y^{12} \]
Step 2: Rewrite the expression
Now the original expression becomes: \[ \frac{16x^8y^{12}}{2yx^3} \cdot (2xy^3) \]
Step 3: Simplify the division
Next, let's simplify \(\frac{16x^8y^{12}}{2yx^3}\):
- Divide \(16\) by \(2\): \[ \frac{16}{2} = 8 \]
- Apply the quotient rule for exponents for \(x\) and \(y\): \[ \frac{x^8}{x^3} = x^{8-3} = x^5 \] \[ \frac{y^{12}}{y} = y^{12-1} = y^{11} \]
So the division simplifies to: \[ 8x^5y^{11} \]
Step 4: Now, multiply by the remaining term
Now we multiply by \(2xy^3\): \[ (8x^5y^{11}) \cdot (2xy^3) = 8 \cdot 2 \cdot x^5 \cdot x \cdot y^{11} \cdot y^3 \] Combine the constants, and apply the product rule for exponents: \[ 16 \cdot x^{5+1} \cdot y^{11+3} = 16x^6y^{14} \]
Final Answer
Thus, the final simplified expression is: \[ \boxed{16x^6y^{14}} \]