1. **State the problem:** Simplify the expression $$\sqrt{(3y^2)^2} \cdot \left(\sqrt{3y^2}\right)^2$$. 2. **Recall the properties of square roots and exponents:** - The square root of a square cancels out: $$\sqrt{a^2} = |a|$$. - The square of a square root cancels out: $$\left(\sqrt{a}\right)^2 = a$$. - Multiplication is associative and commutative. 3. **Simplify each part:** - First term: $$\sqrt{(3y^2)^2} = |3y^2| = 3y^2$$ (assuming $$y^2 \geq 0$$ so absolute value is $$3y^2$$). - Second term: $$\left(\sqrt{3y^2}\right)^2 = 3y^2$$. 4. **Multiply the simplified terms:** $$3y^2 \cdot 3y^2 = 9y^4$$. 5. **Final answer:** $$\boxed{9y^4}$$