https://file.aiquickdraw.com/mathbot/file/17255058592493jwhyj90.jpg

The image asks to arrange the given expressions from least to greatest. Let's break down the expressions first:

1. \( \frac{2^5}{10} \)
2. \( \left( \frac{4^3}{5^2} \right)^2 \)
3. \( \frac{10^7}{10^4} \)
4. \( \left( \frac{2^3}{5^4} \right)^4 \)

### Step-by-step Evaluation:

1. **\( \frac{2^5}{10} \)**:
   - \( 2^5 = 32 \)
   - Therefore, \( \frac{2^5}{10} = \frac{32}{10} = 3.2 \)

2. **\( \left( \frac{4^3}{5^2} \right)^2 \)**:
   - \( 4^3 = 64 \), and \( 5^2 = 25 \)
   - So, \( \frac{64}{25} \approx 2.56 \)
   - Now square it: \( (2.56)^2 = 6.5536 \)

3. **\( \frac{10^7}{10^4} \)**:
   - Simplifying, \( \frac{10^7}{10^4} = 10^{7-4} = 10^3 = 1000 \)

4. **\( \left( \frac{2^3}{5^4} \right)^4 \)**:
   - \( 2^3 = 8 \), and \( 5^4 = 625 \)
   - So, \( \frac{8}{625} \approx 0.0128 \)
   - Now raise it to the power of 4: \( (0.0128)^4 \approx 2.68 \times 10^{-8} \)

### Order from Least to Greatest:
- \( \left( \frac{2^3}{5^4} \right)^4 \approx 2.68 \times 10^{-8} \)
- \( \frac{2^5}{10} = 3.2 \)
- \( \left( \frac{4^3}{5^2} \right)^2 \approx 6.5536 \)
- \( \frac{10^7}{10^4} = 1000 \)

So, the order is:
\[
\boxed{\left( \frac{2^3}{5^4} \right)^4, \frac{2^5}{10}, \left( \frac{4^3}{5^2} \right)^2, \frac{10^7}{10^4}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify expressions involving exponents?
2. What are the properties of exponents when dividing powers with the same base?
3. How do you compute fractional powers or roots?
4. What strategies can you use to compare expressions with exponents?
5. How do you work with scientific notation and very small/large numbers?

**Tip**: Always simplify each component of an expression involving powers before performing operations like squaring or raising to higher powers.

Put the expressions in order from least to greatest.

Learn how to compare and arrange exponential expressions in order from least to greatest with step-by-step breakdowns. Suitable for students in Grades 9-11, covering key concepts like exponent rules and properties of powers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation