https://file.aiquickdraw.com/mathbot/file/1727272339652tey1zrsg.jpg

Let's go through the tasks step by step for questions 4, 5, and 6. The instructions for these questions involve simplifying expressions using the laws of square roots (Wurzelgesetze).

### **Task 4**: Simplify using root laws and calculate.

- **(a)** \(\sqrt{2} \cdot \sqrt{32}\)
  
  Simplification:
  \[
  \sqrt{2 \cdot 32} = \sqrt{64} = 8
  \]

- **(b)** \(\sqrt[3]{2} \cdot \sqrt[3]{108}\)
  
  Simplification:
  \[
  \sqrt[3]{2 \cdot 108} = \sqrt[3]{216} = 6
  \]

- **(c)** \(\sqrt{16} \cdot 9\)

  Simplification:
  \[
  \sqrt{144} = 12
  \]

- **(d)** \(\sqrt[3]{125} \cdot 8\)
  
  Simplification:
  \[
  \sqrt[3]{1000} = 10
  \]

- **(e)** \(\frac{\sqrt{2}}{\sqrt{50}}\)
  
  Simplification:
  \[
  \frac{\sqrt{2}}{\sqrt{50}} = \sqrt{\frac{2}{50}} = \sqrt{\frac{1}{25}} = \frac{1}{5}
  \]

- **(f)** \(\frac{\sqrt[4]{486}}{\sqrt[4]{6}}\)
  
  Simplification:
  \[
  \sqrt[4]{\frac{486}{6}} = \sqrt[4]{81} = 3
  \]

- **(g)** \(\sqrt[4]{20} \cdot \sqrt[4]{500}\)
  
  Simplification:
  \[
  \sqrt[4]{20 \cdot 500} = \sqrt[4]{10000} = 10
  \]

- **(h)** \(\sqrt[3]{\frac{27}{64}}\)
  
  Simplification:
  \[
  \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4}
  \]

- **(i)** Missing

- **(j)** \(\sqrt[5]{1024 \cdot 243}\)
  
  Simplification:
  \[
  \sqrt[5]{1024 \cdot 243} = \sqrt[5]{248832} = 12
  \]

---

### **Task 5**: Write with one root and calculate.

- **(a)** \(\sqrt{\sqrt{256}}\)
  
  Simplification:
  \[
  \sqrt{16} = 4
  \]

- **(b)** \(\sqrt[2]{\sqrt[3]{729}}\)
  
  Simplification:
  \[
  \sqrt[2]{9} = 3
  \]

- **(c)** \(\sqrt[3]{\sqrt{64}}\)
  
  Simplification:
  \[
  \sqrt[3]{8} = 2
  \]

- **(d)** Missing

- **(e)** \(\sqrt[4]{\sqrt[2]{65536}}\)
  
  Simplification:
  \[
  \sqrt[4]{256} = 4
  \]

---

### **Task 6**: Simplify the term using root laws.

- **(a)** \(\sqrt[4]{a} \cdot \sqrt[4]{b}\)
  
  Simplification:
  \[
  \sqrt[4]{ab}
  \]

- **(b)** \(\sqrt[3]{ab^3} \cdot \sqrt{ab}\)
  
  Simplification:
  \[
  ab
  \]

- **(c)** \(\frac{\sqrt[3]{x^6}}{\sqrt[3]{x^9}}\)
  
  Simplification:
  \[
  \frac{x^2}{x^3} = \frac{1}{x}
  \]

- **(d)** \(\sqrt[3]{st} \cdot \sqrt{(st)^3}\)
  
  Simplification:
  \[
  st
  \]

- **(e)** \(\frac{\sqrt[2]{50x} \cdot \sqrt[3]{9x^3}}{\sqrt[2]{2x^6}}\)
  
  Simplification:
  \[
  \frac{\sqrt{50x} \cdot \sqrt[3]{9x^3}}{\sqrt{2x^6}} = \frac{5x}{x^3} = 1
  \]

- **(f)** \(\sqrt[3]{9x} \cdot \sqrt[3]{3x^2}\)
  
  Simplification:
  \[
  \sqrt[3]{27x^3} = 3x
  \]

- **(g)** \(\frac{\sqrt{(x+6)^2}}{\sqrt{12^2 - 36}}\)
  
  Simplification:
  \[
  \frac{x+6}{6}
  \]

- **(h)** \(\frac{\sqrt[2]{a^4b^4}}{(ab)^2}\)
  
  Simplification:
  \[
  1
  \]

- **(i)** \(\sqrt{(ab)^4}\)
  
  Simplification:
  \[
  (ab)^2
  \]

- **(j)** \(\sqrt[2]{\sqrt[3]{a^2}}\)
  
  Simplification:
  \[
  \sqrt[6]{a^2}
  \]

---

Let me know if you'd like more details or have any questions. 

Here are some related problems:
1. Simplify \(\frac{\sqrt[3]{8x^3}}{\sqrt[3]{4x}}\).
2. Calculate \(\sqrt[4]{81 \cdot 64}\).
3. Solve \(\sqrt[5]{243x^5}\).
4. Find the value of \(\frac{\sqrt[3]{1000}}{\sqrt[2]{10^2}}\).
5. Simplify \(\frac{\sqrt{x+7}}{\sqrt[3]{x-1}}\).

**Tip**: Always check whether terms inside a radical can be factored for easier simplification.

Learn how to simplify radical expressions using root laws (Wurzelgesetze) in tasks 4, 5, and 6. This covers key mathematical concepts such as exponents, root simplification, and calculating cube roots and fourth roots. Suitable for students in Grades 9-10.

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