1. **State the problem:** We have triangle ABC with M as the midpoint of AC. Given vectors \(\overrightarrow{AB} = 8\mathbf{a} - 4\mathbf{b}\) and \(\overrightarrow{BC} = 10\mathbf{b}\), find \(\overrightarrow{AM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\), fully simplified. 2. **Recall vector addition rules:** \(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\). 3. **Calculate \(\overrightarrow{AC}\):** $$\overrightarrow{AC} = (8\mathbf{a} - 4\mathbf{b}) + 10\mathbf{b} = 8\mathbf{a} + 6\mathbf{b}$$ 4. **Since M is midpoint of AC,** $$\overrightarrow{AM} = \frac{1}{2} \overrightarrow{AC} = \frac{1}{2} (8\mathbf{a} + 6\mathbf{b})$$ 5. **Simplify:** $$\overrightarrow{AM} = 4\mathbf{a} + 3\mathbf{b}$$ **Final answer:** $$\overrightarrow{AM} = 4\mathbf{a} + 3\mathbf{b}$$