https://r2.erweima.ai/mathbot/image/1726904970008hi0ant3s.jpg

Let's break down the problems visible in the image.

### 4. **Express numbers in exponential form with a base of 0.5**  
Convert the following numbers into powers of 0.5:
- \((0.25)^8\)
- \((0.125)^4\)
- \((0.0625)^2\)

### 5. **Simplify the expression**  
You are given an expression to simplify:  
\[
M = (100 - 1) \cdot (100 - 2) \cdot (100 - 3) \cdot \ldots \cdot (100 - 50)
\]

### 6. **Calculate the values**  
Evaluate the following expressions:
- (a) \(\left( \frac{3}{7} \right)^4 \cdot \left( \frac{3}{7} \right)^5 \cdot \left( \frac{3}{7} \right)^7\)
- (b) \(\left( \frac{7}{8} \right)^5 \cdot \left( \frac{7}{8} \right)^4 \cdot \frac{7}{8}\)
- (c) \([0.6]^3 \cdot [0.6]^8\)

### 7. **Evaluate the expressions**  
Calculate the following:
- (a) \(\left( \frac{2}{5} + \frac{1}{2} \right)^2\)
- (b) \(0.75 - \left( \frac{1}{3} \right)^3\)
- (c) \(\left( \frac{3}{5} \right)^5 - (0.36)^5\)
- (d) \(\left( 1 - \frac{1}{3} \right)^4\)

### 8. **Evaluate the expressions**  
Evaluate the following expressions:
- (a) \(\frac{4^3 \cdot 9^7}{27^7 \cdot 8^2}\)
- (b) \(\frac{(-2)^3 \cdot (-2)^7}{3 \cdot 4^6}\)
- (c) \(\frac{(0.2)^5 \cdot (0.09)^3}{(0.2)^7 \cdot (0.3)^4}\)
- (d) \(\frac{23 + 24 + 25}{7^2}\)

### 9. **Physics problem on planetary masses and distances**  
- (a) Earth’s mass is \(5.97 \times 10^{24} \, \text{kg}\), the Moon’s mass is \(7.35 \times 10^{22} \, \text{kg}\). Calculate the combined mass.
- (b) Compare distances of planets from Earth:  
   - Jupiter is \(8.27 \times 10^8 \, \text{km}\)
   - Uranus is \(3.09 \times 10^9 \, \text{km}\)  
   Determine which is closer.

---

Would you like to start solving specific questions from this set? Let me know which one you'd prefer to work on first!

### 5 Related Questions:
1. How do you convert decimal fractions like 0.25 into powers of 0.5?
2. What is the pattern in the sequence given in problem 5?
3. How do you simplify expressions involving powers of fractions?
4. How do you solve expressions involving mixed operations of addition and exponents?
5. What are the key steps in comparing distances between celestial bodies?

**Tip:** When simplifying expressions, factor powers first before handling coefficients to make the process smoother!

Solve the following math problems involving exponentiation, simplification of expressions, and physics-based calculations related to planetary mass and distance.

Learn how to solve exponentiation problems, simplify fractions, and calculate planetary mass and distance in this detailed math and physics exercise. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation