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You've uploaded a math exercise involving the calculation of various roots and exponents. Here are the problems listed and their solutions:

1. \(\sqrt{81} \div \sqrt{(-5)^2}\)
   - \(\sqrt{81} = 9\) and \(\sqrt{(-5)^2} = \sqrt{25} = 5\)
   - Result: \(9 \div 5 = 1.8\)

2. \(\sqrt{\frac{112^2}{49}} \div \sqrt{32}\)
   - \(\frac{112^2}{49} = \frac{12544}{49} = 256\) and \(\sqrt{256} = 16\)
   - \(\sqrt{32} = 4 \sqrt{2}\)
   - Result: \(\frac{16}{4 \sqrt{2}} = \frac{4}{\sqrt{2}} = 2 \sqrt{2}\)

3. \(\sqrt[3]{\frac{312}{16}} \div \sqrt{\sqrt{16}}\)
   - \(\frac{312}{16} = 19.5\) and \(\sqrt[3]{19.5} \approx 2.699\)
   - \(\sqrt{\sqrt{16}} = \sqrt{4} = 2\)
   - Result: \(\frac{2.699}{2} \approx 1.3495\)

4. \(\sqrt[3]{27} \div \sqrt[3]{3}\)
   - \(\sqrt[3]{27} = 3\) and \(\sqrt[3]{3} = 1.44225\)
   - Result: \(3 \div 1.44225 \approx 2.08\)

These solutions provide a step-by-step calculation for each of the nested and complex root operations shown in the exercise. If you have any questions or need further explanations on any step, please let me know.

### Additional Questions:
1. How do we rationalize a denominator involving square roots?
2. What is the general method for finding cube roots without a calculator?
3. Why does squaring a negative number result in a positive number?
4. What are the properties of square roots that simplify calculations in nested roots?
5. Can you explain how to simplify higher indices of roots, like fourth roots or fifth roots?

### Tip:
Always simplify the terms inside the roots or exponents as much as possible before proceeding with the calculations to make the steps easier and more manageable.

Calculez les racines carrées et cubiques de l'exercice fourni dans l'image.

Solve various problems involving square and cube roots from the exercise. This includes calculating square roots like √81, cube roots like ∛27, and handling nested radical expressions. Suitable for students in Grades 9-11.

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