1. **Problem:** Simplify $\sqrt{12x^{2}y}$. 2. **Formula and rules:** The square root of a product is the product of the square roots: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$. Also, $\sqrt{x^{2}} = |x|$, but we assume $x \geq 0$ for simplicity. 3. **Work:** $$\sqrt{12x^{2}y} = \sqrt{12} \times \sqrt{x^{2}} \times \sqrt{y}$$ 4. Factor 12 into $4 \times 3$: $$= \sqrt{4} \times \sqrt{3} \times x \times \sqrt{y}$$ 5. Simplify square roots: $$= 2x \sqrt{3y}$$ --- 1. **Problem:** Simplify $\sqrt[3]{-27x^{6}}$. 2. **Formula and rules:** Cube root of a product is product of cube roots: $\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}$. Also, $\sqrt[3]{x^{3}} = x$. 3. **Work:** $$\sqrt[3]{-27x^{6}} = \sqrt[3]{-27} \times \sqrt[3]{x^{6}}$$ 4. Simplify: $$= -3 \times x^{2} = -3x^{2}$$ --- 1. **Problem:** Express $\sqrt{-100}$ in terms of $i$. 2. **Formula and rules:** $\sqrt{-a} = i\sqrt{a}$ for $a > 0$. 3. **Work:** $$\sqrt{-100} = i \sqrt{100} = 10i$$ --- 1. **Problem:** Compute $(6 + 6i) - (-4 + i)$. 2. **Formula and rules:** Subtract complex numbers by subtracting real and imaginary parts separately. 3. **Work:** $$(6 + 6i) - (-4 + i) = 6 + 6i + 4 - i = (6 + 4) + (6i - i) = 10 + 5i$$