1. **Problem a:** Simplify $$\left(\frac{a^{2}}{3} \times \left[\frac{a^{3}}{1} \times \frac{1}{a^{4}}\right]\right)^{6 \times \frac{1}{7}}$$. 2. First, simplify inside the brackets: $$\frac{a^{3}}{1} \times \frac{1}{a^{4}} = a^{3} \times a^{-4} = a^{3-4} = a^{-1}$$ 3. Now the expression inside the parentheses is: $$\frac{a^{2}}{3} \times a^{-1} = \frac{a^{2} \times a^{-1}}{3} = \frac{a^{2-1}}{3} = \frac{a^{1}}{3} = \frac{a}{3}$$ 4. The exponent outside is: $$6 \times \frac{1}{7} = \frac{6}{7}$$ 5. So the expression becomes: $$\left(\frac{a}{3}\right)^{\frac{6}{7}} = \frac{a^{\frac{6}{7}}}{3^{\frac{6}{7}}}$$ --- 1. **Problem b:** Simplify $$\left(3 \times \sqrt{\frac{27}{8}}\right)^{2}$$. 2. Simplify inside the square root: $$\sqrt{\frac{27}{8}} = \frac{\sqrt{27}}{\sqrt{8}} = \frac{3\sqrt{3}}{2\sqrt{2}}$$ 3. Multiply by 3: $$3 \times \frac{3\sqrt{3}}{2\sqrt{2}} = \frac{9\sqrt{3}}{2\sqrt{2}}$$ 4. Square the entire expression: $$\left(\frac{9\sqrt{3}}{2\sqrt{2}}\right)^{2} = \frac{81 \times 3}{4 \times 2} = \frac{243}{8}$$ --- 1. **Problem c:** Simplify $$\frac{5^{m} 5^{2} + 5^{m} 5^{1}}{5^{m} 5^{1}}$$. 2. Use exponent addition: $$5^{m} 5^{2} = 5^{m+2}, \quad 5^{m} 5^{1} = 5^{m+1}$$ 3. Rewrite numerator: $$5^{m+2} + 5^{m+1}$$ 4. Factor out $$5^{m+1}$$: $$5^{m+1}(5^{1} + 1) = 5^{m+1}(5 + 1) = 5^{m+1} \times 6$$ 5. Denominator is $$5^{m+1}$$. 6. Simplify fraction: $$\frac{5^{m+1} \times 6}{5^{m+1}} = \cancel{5^{m+1}} \times 6 / \cancel{5^{m+1}} = 6$$ **Final answers:** - a) $$\frac{a^{\frac{6}{7}}}{3^{\frac{6}{7}}}$$ - b) $$\frac{243}{8}$$ - c) $$6$$