1. Problem: Find the value of $|3\mathbf{v} + \mathbf{w}|$ where $\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\mathbf{w} = 5\mathbf{i} - \mathbf{j} + 3\mathbf{k}$. 2. Formula: The magnitude of a vector $\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}$ is given by $$|\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}.$$ 3. Calculate $3\mathbf{v}$: $$3\mathbf{v} = 3(3\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) = 9\mathbf{i} - 6\mathbf{j} + 6\mathbf{k}.$$ 4. Calculate $3\mathbf{v} + \mathbf{w}$: $$3\mathbf{v} + \mathbf{w} = (9 + 5)\mathbf{i} + (-6 - 1)\mathbf{j} + (6 + 3)\mathbf{k} = 14\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}.$$ 5. Find the magnitude: $$|3\mathbf{v} + \mathbf{w}| = \sqrt{14^2 + (-7)^2 + 9^2} = \sqrt{196 + 49 + 81} = \sqrt{326}.$$ 6. Simplify the square root if possible: $$\sqrt{326} = \sqrt{2 \times 163}.$$ Since 163 is prime, this is the simplest form. 7. Final answer: $$|3\mathbf{v} + \mathbf{w}| = \sqrt{326}.$$