1. Evaluate the expression $$\frac{11}{22}x^{-22} + \frac{11}{33}x^{-33} + \frac{11}{22}x^{-44}$$ for given options. Step 1: Simplify each term. $$\frac{11}{22} = \frac{1}{2}, \quad \frac{11}{33} = \frac{1}{3}$$ Step 2: Expression becomes: $$\frac{1}{2}x^{-22} + \frac{1}{3}x^{-33} + \frac{1}{2}x^{-44}$$ Step 3: Without a specific value for $x$, we cannot simplify further. Assuming $x=1$, sum is: $$\frac{1}{2} + \frac{1}{3} + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} + \frac{1}{3} = 1 + \frac{1}{3} = \frac{4}{3}$$ None of the options match this, so more context is needed. 2. Convert 0.000000078 to standard form. Step 1: Count decimal places to move decimal point right to get a number between 1 and 10. $$0.000000078 = 7.8 \times 10^{-8}$$ Answer: (a) 7.8× 10^{-8} 3. Find multiplicative inverse of $$\frac{77}{99}x^{-33}$$. Step 1: Multiplicative inverse is reciprocal: $$\left(\frac{77}{99}x^{-33}\right)^{-1} = \frac{99}{77}x^{33}$$ Answer: (c) $$\frac{99}{77}x^{33}$$ 4. Assertion: $$2^{-3} = \frac{1}{2^3}$$ is true. Reason: Negative exponent means reciprocal of base raised to positive exponent, also true. Reason correctly explains Assertion. Answer: (a) 5. Assertion: Multiplicative inverse of 10^5 is 10^{-5} is false. Reason: Multiplicative inverse of a number multiplied by original number gives 1 is true. Answer: (d) 6. Evaluate $$(5^{-1} \times 2^{-1}) \times 6^{-1}$$. Step 1: Write as reciprocals: $$\frac{1}{5} \times \frac{1}{2} \times \frac{1}{6} = \frac{1}{60}$$ 7. Express in standard form: a) 0.00001275 m = $$1.275 \times 10^{-5}$$ m b) 0.00000000000000000016 coulomb = $$1.6 \times 10^{-19}$$ coulomb 8. Find $m$ such that: $$( -3 )^{m(m+1)} \times ( -3 )^{5} = ( -3 )^{7}$$ Step 1: Add exponents: $$m(m+1) + 5 = 7$$ Step 2: Solve quadratic: $$m^2 + m + 5 = 7 \Rightarrow m^2 + m - 2 = 0$$ Step 3: Factor: $$(m+2)(m-1) = 0 \Rightarrow m = -2 \text{ or } 1$$ 9. Population ratio: $$\frac{1.8 \times 10^{6}}{9 \times 10^{3}} = \frac{1.8}{9} \times 10^{6-3} = 0.2 \times 10^{3} = 200$$ Population in 2010 is 200 times that in 2012. 10. Total viewers: $$8.5 \times 10^{6} + 1.15 \times 10^{6} = (8.5 + 1.15) \times 10^{6} = 9.65 \times 10^{6}$$ 11. Simplify and write in positive exponents: a) $$\frac{2^{x} x^{-3}}{x^{-8} x^{4}} = 2^{x} x^{-3 - (-8) - 4} = 2^{x} x^{1} = 2^{x} x$$ b) $$6^{m} m^{5} \times 3^{m} m^{-8} = (6 \times 3)^{m} m^{5 - 8} = 18^{m} m^{-3}$$ c) $$\frac{x^{x} x^{8}}{x^{3} 4} = \frac{x^{x+8}}{4 x^{3}} = \frac{x^{x+5}}{4}$$ d) $$-3^{4} x^{x} x^{-1} = -81 x^{x-1}$$ e) $$x^{x^{6}} x^{x^{4}} = x^{x^{6} + x^{4}}$$ 12. Simplify: $$\frac{25 \times t^{t^{-4}}}{5^{-3} \times 125 \times t^{t^{-8}}}$$ Step 1: Simplify constants: $$25 = 5^{2}, \quad 125 = 5^{3}, \quad 5^{-3} = \frac{1}{5^{3}}$$ Step 2: Substitute: $$\frac{5^{2} t^{t^{-4}}}{\frac{1}{5^{3}} \times 5^{3} t^{t^{-8}}} = \frac{5^{2} t^{t^{-4}}}{t^{t^{-8}}} = 5^{2} t^{t^{-4} - t^{-8}}$$ Case Study: a) Distance of Mercury from Sun in standard form: $$5.79 \times 10^{7} \text{ km}$$ b) Sum of distances of Mercury and Venus: $$5.79 \times 10^{7} + 1.082 \times 10^{8} = 1.661 \times 10^{8} \text{ km}$$ c) Difference of distances of Earth and Venus: $$1.496 \times 10^{8} - 1.082 \times 10^{8} = 4.14 \times 10^{7} \text{ km}$$