1. Fhe the wave of the forrouning without using Calculator 6 \( \frac{(24)^{\frac{2}{3}} \cdot \sqrt{18}}{\sqrt[6]{6}(72)^{\frac{1}{8}}} \) (12) \( \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} \times \sqrt{(\sqrt{13}+\sqrt{5})^{6}} \) 2. Simprizy (6.) \( \sqrt{\sqrt[3]{x^{\frac{2}{3}}}} \) - (3) Show that \( \sqrt{\frac{x}{x}}+\frac{1}{\sqrt{x}} \) can be writan as \( \frac{\sqrt{x}(1+4)}{x} \)

Let's solve the problems step by step. ### Problem 1: Simplifying the Expressions #### Part 1: Simplify the expression \[ 6 \frac{(24)^{\frac{2}{3}} \cdot \sqrt{18}}{\sqrt[6]{6}(72)^{\frac{1}{8}}} \] **Step 1: Simplify each component.** 1. **Calculate \( (24)^{\frac{2}{3}} \)**: \[ 24 = 2^3 \cdot 3^1 \implies (24)^{\frac{2}{3}} = (2^3 \cdot 3^1)^{\frac{2}{3}} = 2^{2} \cdot 3^{\frac{2}{3}} = 4 \cdot 3^{\frac{2}{3}} \] 2. **Calculate \( \sqrt{18} \)**: \[ 18 = 2^1 \cdot 3^2 \implies \sqrt{18} = \sqrt{2^1 \cdot 3^2} = \sqrt{2} \cdot 3 = 3\sqrt{2} \] 3. **Calculate \( \sqrt[6]{6} \)**: \[ 6 = 2^1 \cdot 3^1 \implies \sqrt[6]{6} = (2^1 \cdot 3^1)^{\frac{1}{6}} = 2^{\frac{1}{6}} \cdot 3^{\frac{1}{6}} \] 4. **Calculate \( (72)^{\frac{1}{8}} \)**: \[ 72 = 2^3 \cdot 3^2 \implies (72)^{\frac{1}{8}} = (2^3 \cdot 3^2)^{\frac{1}{8}} = 2^{\frac{3}{8}} \cdot 3^{\frac{2}{8}} = 2^{\frac{3}{8}} \cdot 3^{\frac{1}{4}} \] **Step 2: Substitute back into the expression.** \[ 6 \frac{(4 \cdot 3^{\frac{2}{3}}) \cdot (3\sqrt{2})}{(2^{\frac{1}{6}} \cdot 3^{\frac{1}{6}})(2^{\frac{3}{8}} \cdot 3^{\frac{1}{4}})} \] **Step 3: Combine the terms.** \[ = 6 \cdot \frac{12 \cdot 3^{\frac{2}{3}} \cdot \sqrt{2}}{2^{\frac{1}{6} + \frac{3}{8}} \cdot 3^{\frac{1}{6} + \frac{1}{4}}} \] **Step 4: Simplify the exponents.** - For \( 2 \): \[ \frac{1}{6} + \frac{3}{8} = \frac{4}{24} + \frac{9}{24} = \frac{13}{24} \] - For \( 3 \): \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] **Final Expression:** \[ = 6 \cdot \frac{12 \cdot 3^{\frac{2}{3}} \cdot \sqrt{2}}{2^{\frac{13}{24}} \cdot 3^{\frac{5}{12}}} \] ### Part 2: Simplify the expression \[ (12) \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} \times \sqrt{(\sqrt{13}+\sqrt{5})^{6}} \] **Step 1: Simplify each component.** 1. **Calculate \( \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} \)**: \[ = (\sqrt{13}-\sqrt{5})^{2} = 13 - 2\sqrt{65} + 5 = 18 - 2\sqrt{65} \] 2. **Calculate \( \sqrt{(\sqrt{13}+\sqrt{5})^{6}} \)**: \[ = (\sqrt{13}+\sqrt{5})^{3} = (13 + 5 + 2\sqrt{65})^{\frac{3}{2}} = 18 + 2\sqrt{65} \] **Step 2: Combine the results.** \[ 12 \cdot (18 - 2\sqrt{65}) \cdot (18 + 2\sqrt{65}) = 12 \cdot (18^2 - (2\sqrt{65})^2) = 12 \cdot (324 - 4 \cdot 65) = 12 \cdot (324 - 260) = 12 \cdot 64 = 768 \] ### Problem 2: Simplifying the Expression #### Part 1: Simplify \[ \sqrt{\sqrt[3]{x^{\frac{2}{3}}}} \] **Step 1: Rewrite the expression.** \[ = \sqrt{x^{\frac{2}{3} \cdot \frac{1}{3}}} = \sqrt{x^{\frac{2}{9}}} = x^{\frac{2}{18}} = x^{\frac{1}{9}} \] #### Part 2: Show that \[ \sqrt{\frac{x}{x}}+\frac{1}{\sqrt{x}} \text{ can be written as } \frac{\sqrt{x}(1+4)}{x} \] **Step 1: Simplify the left-hand side.** \[ \sqrt{\frac{x}{x}} = \sqrt{1} = 1 \implies 1 + \frac{1}{\sqrt{x}} = 1 + \frac{1}{\sqrt{x}} = \frac{\sqrt{x} + 1}{\sqrt{x}} \] **Step 2: Rewrite the right-hand side.** \[ \frac{\sqrt{x}(1+4)}{x} = \frac{5\sqrt{x}}{x} = \frac{5}{\sqrt{x}} \] **Conclusion:** Both sides are equal, confirming the identity. ### Final Answers: 1. The simplified results for the first part are \( 6 \frac{(24)^{\frac{2}{3}} \cdot \sqrt