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The problem is to simplify the expression $((-2xy^2)^2) (\frac{x^6}{(2x)^2})^3$.

AlgebraExponentsSimplificationAlgebraic Expressions
2025/6/5

1. Problem Description

The problem is to simplify the expression ((−2xy2)2)(x6(2x)2)3((-2xy^2)^2) (\frac{x^6}{(2x)^2})^3.

2. Solution Steps

First, simplify the term (−2xy2)2(-2xy^2)^2.
(−2xy2)2=(−2)2x2(y2)2=4x2y4(-2xy^2)^2 = (-2)^2 x^2 (y^2)^2 = 4x^2y^4
Next, simplify the term (x6(2x)2)3(\frac{x^6}{(2x)^2})^3.
(x6(2x)2)3=(x64x2)3=(14x6−2)3=(14x4)3=(14)3(x4)3=164x12(\frac{x^6}{(2x)^2})^3 = (\frac{x^6}{4x^2})^3 = (\frac{1}{4}x^{6-2})^3 = (\frac{1}{4}x^4)^3 = (\frac{1}{4})^3 (x^4)^3 = \frac{1}{64}x^{12}
Now, multiply the two simplified terms:
(4x2y4)(164x12)=464x2+12y4=116x14y4(4x^2y^4) (\frac{1}{64}x^{12}) = \frac{4}{64} x^{2+12} y^4 = \frac{1}{16}x^{14}y^4

3. Final Answer

The final answer is 116x14y4\frac{1}{16}x^{14}y^4.

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