https://r2.erweima.ai/mathbot/image/17271944357887lc6xoyj.jpg

Let's work through each problem in the image you provided.

### 1. Beantworte die Aussagen:
**a) Gibt es Zahlen, die sowohl irrational als auch rational sind?**  
- **Answer:** No. A number cannot be both irrational and rational by definition. A rational number can be expressed as a fraction of two integers, while an irrational number cannot.

**b) Ist \( \sqrt{144} \) eine irrationale Zahl?**  
- **Answer:** No, \( \sqrt{144} = 12 \), which is a rational number.

**c) Ist die Zahl 1.34343434444... eine rationale Zahl?**  
- **Answer:** No, this number is non-repeating and non-terminating, which means it is irrational.

**d) Ist \( \sqrt{81} \) eine reelle Zahl?**  
- **Answer:** Yes, \( \sqrt{81} = 9 \), and 9 is a real number.

---

### 2. Ein Würfel hat den Oberflächeninhalt 150 cm². Bestimme die Kantenlänge des Würfels.
- The surface area \( A \) of a cube is given by \( A = 6a^2 \), where \( a \) is the side length.
  - We have \( A = 150 \, \text{cm}^2 \), so:
  \[
  6a^2 = 150
  \]
  \[
  a^2 = \frac{150}{6} = 25
  \]
  \[
  a = \sqrt{25} = 5 \, \text{cm}
  \]
  - **Answer:** The side length is 5 cm.

---

### 3. Löse die folgenden Gleichungen. Eine hat keine Lösung:
**a) \( x^2 + 7 = 88 \)**  
\[
x^2 = 88 - 7 = 81
\]
\[
x = \pm \sqrt{81} = \pm 9
\]

**b) \( 5x^2 + 9 = 54 \)**  
\[
5x^2 = 54 - 9 = 45
\]
\[
x^2 = \frac{45}{5} = 9
\]
\[
x = \pm \sqrt{9} = \pm 3
\]

**c) \( 9x^2 - 9 = -27 \)**  
\[
9x^2 = -27 + 9 = -18
\]
\[
x^2 = \frac{-18}{9} = -2
\]
  - There is no real solution for this equation because you cannot take the square root of a negative number in the real number system.

**d) \( -x^2 + 10 = 5 \)**  
\[
-x^2 = 5 - 10 = -5
\]
\[
x^2 = 5
\]
\[
x = \pm \sqrt{5}
\]

**e) \( 2x^2 = x^2 \)**  
\[
2x^2 - x^2 = 0
\]
\[
x^2 = 0 \quad \Rightarrow \quad x = 0
\]

**f) \( \sqrt{x^2} = 2x + 3 \)**  
Taking the square root on both sides will complicate the equation, so you must test this for specific values of \( x \) and verify that no solution satisfies the equation.

---

### 4. Ziehe teilweise die Wurzel:
**a) \( \sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30} \)**  
**b) \( \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \)**  
**c) \( \sqrt{\frac{3}{400}} = \frac{\sqrt{3}}{\sqrt{400}} = \frac{\sqrt{3}}{20} \)**  
**d) \( \sqrt{0.36} = \sqrt{\frac{36}{100}} = \frac{6}{10} = 0.6 \)**

---

### 5. Bringe den Vorfaktor unter das Wurzelzeichen:
**a) \( 2 \cdot \sqrt{17} = \sqrt{4 \cdot 17} = \sqrt{68} \)**  
**b) \( 10 \cdot \sqrt{1.33} = \sqrt{100 \cdot 1.33} = \sqrt{133} \)**  
**c) \( \frac{3}{4} \cdot \sqrt{11} = \sqrt{\left( \frac{9}{16} \right) \cdot 11} = \sqrt{\frac{99}{16}} = \frac{\sqrt{99}}{4} \)**  
**d) \( 16 \cdot \sqrt{1} = \sqrt{256 \cdot 1} = \sqrt{256} = 16 \)**

---

### 6. Vereinfache so weit wie möglich:
**a) \( 2\sqrt{5} + 8\sqrt{5} = 10\sqrt{5} \)**  
**b) \( 5\sqrt{7} - 9\sqrt{7} = -4\sqrt{7} \)**  
**c) \( \sqrt{45} + \sqrt{36} = \sqrt{9 \cdot 5} + 6 = 3\sqrt{5} + 6 \)**  

---

Would you like further clarifications on any of the problems, or should I go into more detail on any specific topic?

Here are 5 additional questions to deepen your understanding:

1. What is the difference between rational and irrational numbers?
2. How do you solve quadratic equations with no real solutions?
3. Can you think of another method to simplify \( \sqrt{72} \) without splitting into factors?
4. Why is the cube's surface area formula \( 6a^2 \)?
5. How do you know if a number has no real square root?

**Tip:** When solving square roots, always check if you can break the number into perfect squares for simplification.

Series of math problems including rational and irrational numbers, surface area of a cube, quadratic equations, and simplification.

This problem set includes solving quadratic equations, understanding rational and irrational numbers, calculating the surface area of a cube, and simplifying square roots. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation