https://r2.erweima.ai/mathbot/image/1725547833117245kjgzi.jpg

From the image you provided, I see multiple mathematical expressions, many of which involve exponents. Let's break them down and solve each part:

1. **First expression**:  
   \[
   \frac{5^{-3}}{5^3}
   \]
   Using the law of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we get:
   \[
   5^{-3-3} = 5^{-6}
   \]

2. **Second expression**:  
   \[
   (8 - 2 + (1 + 8) - 5)^2
   \]
   Simplify the terms inside the parentheses:
   \[
   (8 - 2 + 9 - 5)^2 = (6 + 9 - 5)^2 = 10^2 = 100
   \]

3. **Third expression**:  
   \[
   m^6 \cdot m^3
   \]
   Use the product of powers rule \(a^m \cdot a^n = a^{m+n}\):
   \[
   m^{6+3} = m^9
   \]

4. **Fourth expression**:  
   \[
   \left(\frac{4^6}{4}\right) \cdot (3^3 \cdot 3^2)
   \]
   Simplify each part:
   - \(\frac{4^6}{4} = 4^{6-1} = 4^5\)
   - \(3^3 \cdot 3^2 = 3^{3+2} = 3^5\)

   So the simplified expression becomes:
   \[
   4^5 \cdot 3^5 = (4 \cdot 3)^5 = 12^5
   \]

5. **Fifth expression**:  
   \[
   (5 \cdot 8)^2 - (2 \cdot 2 \cdot 10)^0
   \]
   Simplify:
   - \( (5 \cdot 8)^2 = 40^2 = 1600 \)
   - \( (2 \cdot 2 \cdot 10)^0 = 1 \) (since any non-zero number raised to the power of 0 is 1)

   So the final result is:
   \[
   1600 - 1 = 1599
   \]

### Final Results:
1. \( \frac{5^{-3}}{5^3} = 5^{-6} \)
2. \( (8 - 2 + (1 + 8) - 5)^2 = 100 \)
3. \( m^6 \cdot m^3 = m^9 \)
4. \( \left(\frac{4^6}{4}\right) \cdot (3^3 \cdot 3^2) = 12^5 \)
5. \( (5 \cdot 8)^2 - (2 \cdot 2 \cdot 10)^0 = 1599 \)

Would you like more details on any of these steps, or have any further questions?

Here are some related questions for practice:
1. Simplify \(\frac{2^5}{2^3}\).
2. Solve \((7 + 3 - 5 + 4)^2\).
3. What is the result of \(x^4 \cdot x^5\)?
4. Simplify \(\left(\frac{3^4}{3}\right) \cdot (2^3 \cdot 2^2)\).
5. Find the value of \((6 \cdot 7)^2 - (3 \cdot 3 \cdot 2)^0\).

**Tip**: When simplifying expressions with exponents, remember that multiplying terms with the same base adds their exponents, while dividing subtracts their exponents.

Solve the following mathematical expressions involving exponents:
1. 5^(-3) / 5^3
2. (8 - 2 + (1 + 8) - 5)^2
3. m^6 * m^3
4. (4^6 / 4) * (3^3 * 3^2)
5. (5 * 8)^2 - (2 * 2 * 10)^0

This page provides a detailed solution to five exponent-related problems, including the division and multiplication of powers, and the zero exponent rule. Learn how to simplify expressions such as 5^(-3) / 5^3, m^6 * m^3, and others, 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