https://r2.erweima.ai/mathbot/image/17267866579393p0yx6w6.jpg

Let's start by solving the problems in the image you uploaded. The image contains two sets of problems: one with exponentiation and one requiring conversion to scientific notation. Here's a breakdown of the solutions:

### **Exponentiation Problems:**

1. **\(3^4 \cdot 3^5\)**
   - Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
   \[
   3^4 \cdot 3^5 = 3^{4+5} = 3^9 = 19683
   \]
   
2. **\((5^6)^3\)**
   - Using the power of a power rule: \((a^m)^n = a^{m \cdot n}\):
   \[
   (5^6)^3 = 5^{6 \cdot 3} = 5^{18}
   \]
   - \(5^{18}\) is a very large number: 
   \[
   5^{18} = 3814697265625
   \]

3. **\(\frac{9^3}{9^2}\)**
   - Using the division rule for exponents: \(\frac{a^m}{a^n} = a^{m-n}\):
   \[
   \frac{9^3}{9^2} = 9^{3-2} = 9^1 = 9
   \]

4. **\(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\)**
   - This is equivalent to \(5^7\):
   \[
   5^7 = 78125
   \]

5. **\(13 \times 13 \times 13 \times 13 \times 13\)**
   - This is equivalent to \(13^5\):
   \[
   13^5 = 371293
   \]

### **Scientific Notation Problems:**

1. **\(13800000000000\)**
   - Convert to scientific notation: 
   \[
   1.38 \times 10^{13}
   \]

2. **\(0.000000057\)**
   - Convert to scientific notation:
   \[
   5.7 \times 10^{-8}
   \]

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Would you like more detailed steps on any of these, or do you have any other questions?

Here are 5 related questions for practice:
1. Simplify \(7^4 \cdot 7^2\).
2. Express \((4^3)^2\) in its simplest form.
3. Convert \(4500000\) into scientific notation.
4. Simplify \(\frac{12^6}{12^3}\).
5. Convert \(0.00023\) to scientific notation.

**Tip:** When multiplying powers with the same base, always add the exponents.

Solve the exponentiation and scientific notation problems in the image.

Learn how to solve exponentiation and scientific notation problems step by step. This guide covers laws of exponents, multiplication, division, and conversion to scientific notation. Suitable for students in Grades 8-10.

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