https://r2.erweima.ai/mathbot/image/1723092943420vt1ak04r.jpg

Let's solve the problems presented in the uploaded image step by step.

### 1. Calcula cuando sea posible:

**(a) \(\sqrt[3]{216 \div 6}\)**

First, calculate the division inside the root:
\[ 216 \div 6 = 36 \]
Then, find the cube root:
\[ \sqrt[3]{36} \]

The cube root of 36 is not an integer, so we can leave it in its simplified form:
\[ \sqrt[3]{36} \]

**(b) \(\sqrt[4]{81}\)**

Calculate the fourth root:
\[ \sqrt[4]{81} = \sqrt[4]{3^4} = 3 \]

**(c) \(\sqrt{49}\)**

Calculate the square root:
\[ \sqrt{49} = 7 \]

### 2. Resuelvan de dos formas distintas cuando sea posible:

**(a) \(\sqrt[3]{27 \div 9}\)**

First, calculate the division inside the root:
\[ 27 \div 9 = 3 \]
Then, find the cube root:
\[ \sqrt[3]{3} \]

Alternatively, use properties of roots and division:
\[ \sqrt[3]{27} \div \sqrt[3]{9} = 3 \div \sqrt[3]{9} = 3 \div 2.08 \approx 1.44 \]

**(b) \(\sqrt[5]{25 \div 16}\)**

First, calculate the division inside the root:
\[ 25 \div 16 = 1.5625 \]
Then, find the fifth root:
\[ \sqrt[5]{1.5625} \approx 1.1 \]

Alternatively, using the property of roots:
\[ \sqrt[5]{25} \div \sqrt[5]{16} = 1.9037 \div 1.7411 \approx 1.1 \]

**(c) \(\sqrt{49 + 16 \cdot 4}\)**

First, perform the multiplication inside the root:
\[ 16 \cdot 4 = 64 \]
Then, add inside the root:
\[ 49 + 64 = 113 \]
Finally, find the square root:
\[ \sqrt{113} \approx 10.63 \]

Alternatively, we can directly use the expression and apply the sum:
\[ \sqrt{49 + 64} = \sqrt{113} \approx 10.63 \]

### 3. Resuelvan los siguientes cálculos combinados aplicando propiedades cuando sea posible:

Since the image provided does not contain specific problems for part 3, we'll generalize the approach.

To solve combined calculations involving roots and division:

1. Apply the properties of roots and exponents.
2. Simplify expressions inside the roots before taking the root.

Do you want details or have any questions? Here are some related questions to expand your understanding:

1. How do you calculate cube roots for non-perfect cubes?
2. What are the properties of roots and how can they be used in calculations?
3. How do you simplify expressions inside a radical before solving?
4. Can roots be negative? Explain with an example.
5. What is the general method for solving combined calculations involving roots and exponents?
6. How do you handle division inside a root when the numbers are large?
7. What are the differences between square roots and cube roots in terms of calculation?
8. How can you approximate roots for non-perfect squares or cubes?

**Tip:** When solving roots, always check if the expression inside can be simplified first for easier calculation.

Learn how to solve root and radical problems step-by-step with detailed explanations and examples. This includes calculating cube roots, fourth roots, and square roots, and applying properties of roots in calculations. Suitable for Grades 7-9.

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