Math Problem Statement

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

Solution

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

  1. 81÷(5)2\sqrt{81} \div \sqrt{(-5)^2}

    • 81=9\sqrt{81} = 9 and (5)2=25=5\sqrt{(-5)^2} = \sqrt{25} = 5
    • Result: 9÷5=1.89 \div 5 = 1.8
  2. 112249÷32\sqrt{\frac{112^2}{49}} \div \sqrt{32}

    • 112249=1254449=256\frac{112^2}{49} = \frac{12544}{49} = 256 and 256=16\sqrt{256} = 16
    • 32=42\sqrt{32} = 4 \sqrt{2}
    • Result: 1642=42=22\frac{16}{4 \sqrt{2}} = \frac{4}{\sqrt{2}} = 2 \sqrt{2}
  3. 312163÷16\sqrt[3]{\frac{312}{16}} \div \sqrt{\sqrt{16}}

    • 31216=19.5\frac{312}{16} = 19.5 and 19.532.699\sqrt[3]{19.5} \approx 2.699
    • 16=4=2\sqrt{\sqrt{16}} = \sqrt{4} = 2
    • Result: 2.69921.3495\frac{2.699}{2} \approx 1.3495
  4. 273÷33\sqrt[3]{27} \div \sqrt[3]{3}

    • 273=3\sqrt[3]{27} = 3 and 33=1.44225\sqrt[3]{3} = 1.44225
    • Result: 3÷1.442252.083 \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.

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Math Problem Analysis

Mathematical Concepts

Square Roots
Cube Roots
Radicals
Simplification

Formulas

√(a) = x such that x^2 = a
Cube Root: ∛(a) = x such that x^3 = a
Basic properties of exponents and radicals

Theorems

Properties of square roots: √(a^2) = a
Properties of cube roots: ∛(a^3) = a
Rationalization techniques

Suitable Grade Level

Grades 9-11