1. **State the problem:** Evaluate the expression $$a^2 b^5 : c^{-10} + \frac{a - b}{a + b}$$ for $$a=2$$, $$b=4$$, and $$c=-\frac{1}{2}$$. 2. **Recall the rules:** - Division by a negative exponent means multiplication by the positive exponent: $$x^{-n} = \frac{1}{x^n}$$. - The colon symbol ":" means division. - Substitute values carefully. 3. **Rewrite the expression:** $$\frac{a^2 b^5}{c^{-10}} + \frac{a - b}{a + b}$$ 4. **Apply the negative exponent rule:** $$\frac{a^2 b^5}{c^{-10}} = a^2 b^5 \times c^{10}$$ 5. **Substitute values:** $$a=2, b=4, c=-\frac{1}{2}$$ Calculate each part: - $$a^2 = 2^2 = 4$$ - $$b^5 = 4^5 = 1024$$ - $$c^{10} = \left(-\frac{1}{2}\right)^{10} = \left(\frac{1}{2}\right)^{10} = \frac{1}{2^{10}} = \frac{1}{1024}$$ 6. **Calculate the first term:** $$a^2 b^5 c^{10} = 4 \times 1024 \times \frac{1}{1024} = 4 \times \cancel{1024} \times \frac{1}{\cancel{1024}} = 4$$ 7. **Calculate the second term:** $$\frac{a - b}{a + b} = \frac{2 - 4}{2 + 4} = \frac{-2}{6} = -\frac{1}{3}$$ 8. **Add the two terms:** $$4 + \left(-\frac{1}{3}\right) = 4 - \frac{1}{3} = \frac{12}{3} - \frac{1}{3} = \frac{11}{3}$$ **Final answer:** $$\frac{11}{3}$$