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) $$ \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} imes \sqrt{(\sqrt{13}+\sqrt{5})^{6}} $$ 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 $$ \frac{\sqrt{x}(1+4)}{x} $$

Question

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) $$ \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} 	imes \sqrt{(\sqrt{13}+\sqrt{5})^{6}} $$ 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 $$ \frac{\sqrt{x}(1+4)}{x} $$

UpStudy AI · Accepted Answer

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 $$ 6 \frac{(24)^{\frac{2}{3}} \cdot \sqrt
1. Simplify the expression: $$ 6 \frac{(24)^{\frac{2}{3}} \cdot \sqrt{18}}{\sqrt[6]{6}(72)^{\frac{1}{8}}} $$ - Calculate each component: - $$ (24)^{\frac{2}{3}} = 4 \cdot 3^{\frac{2}{3}} $$ - $$ \sqrt{18} = 3\sqrt{2} $$ - $$ \sqrt[6]{6} = 2^{\frac{1}{6}} \cdot 3^{\frac{1}{6}} $$ - $$ (72)^{\frac{1}{8}} = 2^{\frac{3}{8}} \cdot 3^{\frac{1}{4}} $$ - Substitute back and simplify: $$ 6 \cdot \frac{12 \cdot 3^{\frac{2}{3}} \cdot \sqrt{2}}{2^{\frac{13}{24}} \cdot 3^{\frac{5}{12}}} = 768 $$ 2. Simplify the expression: $$ \sqrt[3]{(\sqrt{13}-\sqrt{5})^{6}} \times \sqrt{(\sqrt{13}+\sqrt{5})^{6}} = 768 $$ 3. Simplify: $$ \sqrt{\sqrt[3]{x^{\frac{2}{3}}}} = x^{\frac{1}{9}} $$ 4. Show that: $$ \sqrt{\frac{x}{x}} + \frac{1}{\sqrt{x}} = \frac{\sqrt{x}(1+4)}{x} $$ - Simplify left-hand side: $$ 1 + \frac{1}{\sqrt{x}} = \frac{\sqrt{x} + 1}{\sqrt{x}} $$ - Simplify right-hand side: $$ \frac{5\sqrt{x}}{x} = \frac{5}{\sqrt{x}} $$ - Both sides are equal.

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