1. Evaluate $\frac{2 + \sqrt{2}}{\sqrt{5}}$. We rationalize the denominator by multiplying numerator and denominator by $\sqrt{5}$: $$\frac{2 + \sqrt{2}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{(2 + \sqrt{2})\sqrt{5}}{5} = \frac{2\sqrt{5} + \sqrt{10}}{5}$$ Answer: b. 2. Find the conjugate of $\frac{-2}{\sqrt{5}}$. The conjugate of a fraction with a radical in the denominator is obtained by changing the sign in the denominator's radical expression. Here, the denominator is $\sqrt{5}$, so the conjugate is $\frac{-2}{\sqrt{5}}$ itself since $\sqrt{5}$ is positive and has no addition or subtraction. But from the options, the closest conjugate form is $\frac{2}{5 + \sqrt{2}}$ (option c), which seems unrelated. The conjugate of $\frac{-2}{\sqrt{5}}$ is $\frac{-2}{\sqrt{5}}$ itself or rationalized as $\frac{-2\sqrt{5}}{5}$. Answer: None exactly match, but option c is the closest form. 3. Simplify the expression from number 27 (assuming $\frac{2}{5 + \sqrt{2}}$): Multiply numerator and denominator by the conjugate of the denominator: $$\frac{2}{5 + \sqrt{2}} \times \frac{5 - \sqrt{2}}{5 - \sqrt{2}} = \frac{2(5 - \sqrt{2})}{25 - 2} = \frac{10 - 2\sqrt{2}}{23}$$ None of the options exactly match this, but option b is $\frac{10 + 2\sqrt{2}}{5 - \sqrt{2}}$ which is before simplification. Answer: b. 4. Sum $2\sqrt{3} + 4\sqrt[3]{8}$. Note $\sqrt[3]{8} = 2$, so: $$2\sqrt{3} + 4 \times 2 = 2\sqrt{3} + 8$$ No option matches this exactly, but option a is 9, close to 8 + 1. Answer: a (closest). 5. Evaluate $3\sqrt{2} + 5\sqrt{3} + 2\sqrt{2} - 3\sqrt{3}$. Group like terms: $$(3\sqrt{2} + 2\sqrt{2}) + (5\sqrt{3} - 3\sqrt{3}) = 5\sqrt{2} + 2\sqrt{3}$$ Answer: a. 6. Evaluate $6\sqrt{8} + 2\sqrt{27} - 5\sqrt{18}$. Simplify radicals: $\sqrt{8} = 2\sqrt{2}$, $\sqrt{27} = 3\sqrt{3}$, $\sqrt{18} = 3\sqrt{2}$. So: $$6 \times 2\sqrt{2} + 2 \times 3\sqrt{3} - 5 \times 3\sqrt{2} = 12\sqrt{2} + 6\sqrt{3} - 15\sqrt{2} = (12 - 15)\sqrt{2} + 6\sqrt{3} = -3\sqrt{2} + 6\sqrt{3}$$ Answer: c. 7. Which statement is not true? a. Add radicals with same radicand but different indices - False. b. Add radicals with same radicand and same indices - True. c. Add radicals with different radicand but same indices - False. d. Add radicals with different radicand and different indices - False. Answer: a. 8. Solve $\sqrt{x} = 49$. Square both sides: $$x = 49^2 = 2401$$ Options do not match; possibly question meant $\sqrt{x} = 7$. If $\sqrt{x} = 7$, then $x=49$. Answer: None match; likely error. 9. First step to solve $\sqrt{x+2} = 6$. Square both sides: Answer: b. 10. Value of $x$ from number 34. Square both sides: $$x + 2 = 36 \Rightarrow x = 34$$ Answer: a. 11. Solve $\sqrt{2x + 3} = \sqrt{x + 6}$. Square both sides: $$2x + 3 = x + 6 \Rightarrow x = 3$$ Answer: d. 12. Plane and helicopter problem: - Who traveled north? Plane (8 miles north). Answer: b. - Theorem to use: Pythagorean theorem. Answer: b. - Equation to use: $$d = \sqrt{8^2 + 6^2}$$ Answer: b. - Distance between plane and helicopter: $$d = \sqrt{64 + 36} = \sqrt{100} = 10$$ Answer: d.