https://r2.erweima.ai/mathbot/image/1725882103729rh3k1a8a.jpg

The image you uploaded contains mathematical properties of algebraic expressions with exponents, as well as some examples for simplification. Here's a breakdown of the content:

### Properties of Algebraic Expressions with Exponents:
1. **Power of a number**:  
   \[ x^n = x \cdot x \cdot x \cdot \ldots \cdot x \quad (n \text{ times}) \]

2. **Product of powers with the same base**:  
   \[ x^n \cdot x^k = x^{n+k} \]

3. **Division of powers with the same base**:  
   \[ \frac{x^n}{x^k} = x^{n-k} \quad \text{and} \quad x^0 = 1 \]

4. **Negative exponents**:  
   \[ x^{-k} = \frac{1}{x^k} \quad \text{and} \quad \frac{1}{x^{-k}} = x^k \]

5. **Power of a fraction**:  
   \[ \left(\frac{a}{b}\right)^k = \frac{a^k}{b^k} \quad \text{and} \quad (a \cdot b)^k = a^k \cdot b^k \]

6. **Power of a power**:  
   \[ (x^k)^m = x^{k \cdot m} \]

### Examples for Simplification:
1. \( 2^3 \cdot 6^4 \)
2. \( 2^{3+6^4} \)
3. \( a^3 \cdot 6(ab)^4 \)
4. \( a^3 + (ab)^4 \)
5. \( \frac{x^7}{x^3} \)
6. \( \frac{2^3 \cdot 5^2 + 2^2 \cdot 15}{20} \)
7. \( \frac{2^3 \cdot 3 \cdot 5^2}{2 \cdot 3^2 \cdot 5} \)

Would you like a detailed solution for one or more of these examples? 

---

Here are 5 related questions:
1. How can we simplify \( a^3 \cdot (ab)^4 \)?
2. What happens when the exponent in \( x^n \cdot x^k \) is negative?
3. How would you interpret and simplify the expression \( \frac{x^7}{x^3} \)?
4. What is the rule for simplifying the sum of powers, like \( 2^3 + 2^4 \)?
5. Can you apply the exponent rules to fractions involving variables?

**Tip**: Always group terms with similar bases before applying the exponent rules to simplify expressions efficiently.

Properties of algebraic expressions with exponents and simplification examples, such as 2^3 * 6^4 and (a^3 * (ab)^4).

Learn how to simplify algebraic expressions with exponents using key properties, such as the laws of exponents. Examples include expressions like 2^3 * 6^4 and (a^3 * (ab)^4). Ideal 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