1. **State the problem:** Simplify the expression $4x^6 - \sqrt{4y^2}$. 2. **Recall the formula and rules:** The square root of a square is the absolute value of the base, i.e., $\sqrt{a^2} = |a|$. Also, simplify constants and variables separately. 3. **Simplify the square root term:** $$\sqrt{4y^2} = \sqrt{4} \times \sqrt{y^2} = 2 \times |y| = 2|y|$$ 4. **Rewrite the expression:** $$4x^6 - 2|y|$$ 5. **Final simplified form:** $$4x^6 - 2|y|$$ This is the simplest form unless further information about $y$ is given.