1. Problem: Find the absolute values of the expressions: - (f) $\sqrt{\frac{5}{4}} - \frac{\sqrt{18}}{\sqrt{8}}$ - (g) $\frac{\sqrt{12}}{\sqrt{8}} - 3 + \left(\sqrt{\frac{5}{7}} - 2\right)$ - (h) $\left(\frac{\sqrt{18}}{\sqrt{10}} + \sqrt{14}\right) - (\sqrt{13} - 3)$ 2. Recall the absolute value definition: for any real number $a$, $|a| = a$ if $a \geq 0$, and $|a| = -a$ if $a < 0$. 3. Calculate each expression step-by-step and then take the absolute value. --- **(f)** - Simplify each term: $$\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}$$ $$\frac{\sqrt{18}}{\sqrt{8}} = \sqrt{\frac{18}{8}} = \sqrt{\frac{9 \times 2}{4 \times 2}} = \sqrt{\frac{9}{4}} = \frac{3}{2}$$ - Expression becomes: $$\frac{\sqrt{5}}{2} - \frac{3}{2} = \frac{\sqrt{5} - 3}{2}$$ - Approximate $\sqrt{5} \approx 2.236$: $$\frac{2.236 - 3}{2} = \frac{-0.764}{2} = -0.382$$ - Absolute value: $$| -0.382 | = 0.382$$ --- **(g)** - Simplify each term: $$\frac{\sqrt{12}}{\sqrt{8}} = \sqrt{\frac{12}{8}} = \sqrt{\frac{3}{2}} \approx 1.2247$$ $$\sqrt{\frac{5}{7}} \approx 0.8452$$ - Expression: $$1.2247 - 3 + (0.8452 - 2) = 1.2247 - 3 + 0.8452 - 2 = (1.2247 + 0.8452) - 5 = 2.0699 - 5 = -2.9301$$ - Absolute value: $$| -2.9301 | = 2.9301$$ --- **(h)** - Simplify each term: $$\frac{\sqrt{18}}{\sqrt{10}} = \sqrt{\frac{18}{10}} = \sqrt{1.8} \approx 1.3416$$ $$\sqrt{14} \approx 3.7417$$ $$\sqrt{13} \approx 3.6056$$ - Expression: $$(1.3416 + 3.7417) - (3.6056 - 3) = 5.0833 - 0.6056 = 4.4777$$ - Absolute value: $$|4.4777| = 4.4777$$ --- **Final answers:** - (f) $0.382$ - (g) $2.9301$ - (h) $4.4777$