https://file.aiquickdraw.com/mathbot/file/17273115609195687likb.jpg

Let's go through the problems in the image one by one. The questions involve operations with roots and logarithms. Here's a detailed breakdown of each:

### 1. **Simplify the following expression:**
   \( 3 \cdot 5^{5/2} \)

   This can be expressed as:
   \[
   3 \cdot 5^{5/2} = 3 \cdot \sqrt{5^5} = 3 \cdot 5^{2} \cdot \sqrt{5} = 3 \cdot 25 \cdot \sqrt{5} = 75\sqrt{5}
   \]

   ### 2. **Simplify the following expression:**
   \( 3 \cdot 6^{5/2} \)

   Applying similar logic:
   \[
   3 \cdot 6^{5/2} = 3 \cdot \sqrt{6^5} = 3 \cdot 6^{2} \cdot \sqrt{6} = 3 \cdot 36 \cdot \sqrt{6} = 108\sqrt{6}
   \]

### 3. **Solve the following radical expression:**
   \[
   2 \cdot \sqrt[3]{5} + 3 \cdot \sqrt[3]{5} - \sqrt[3]{5}
   \]

   Combine the terms:
   \[
   (2 + 3 - 1) \cdot \sqrt[3]{5} = 4 \cdot \sqrt[3]{5}
   \]

### 4. **Solve the following expression with roots:**
   \[
   4\sqrt{2} - \sqrt{18} + 2\sqrt{50}
   \]
   Simplify the radicals:
   \[
   \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}, \quad \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
   \]
   Now substitute back:
   \[
   4\sqrt{2} - 3\sqrt{2} + 2 \cdot 5\sqrt{2} = 4\sqrt{2} - 3\sqrt{2} + 10\sqrt{2} = 11\sqrt{2}
   \]

### 5. **Simplify the following division of roots:**
   \[
   \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{3}{5}
   \]

   Multiply the fractions:
   \[
   \frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{3}{5} = \frac{3\sqrt{5}}{5\sqrt{3}} = \frac{\sqrt{15}}{5}
   \]

### 6. **Simplify the following division of radicals:**
   \[
   \frac{2\sqrt{3}}{3\sqrt{5}}
   \]
   This is already simplified, but if we want to rationalize the denominator:
   \[
   \frac{2\sqrt{3}}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{15}}{15}
   \]

### 7. **Logarithmic expression:**
   \( 5 \log 125 \)

   First, express 125 as a power of 5:
   \[
   125 = 5^3
   \]
   Now use logarithmic properties:
   \[
   5 \log 125 = 5 \log (5^3) = 5 \cdot 3 \log 5 = 15 \log 5
   \]

---

Let me know if you'd like any additional steps clarified or further details on any specific part. 

Here are 5 related questions for you:
1. How would you simplify \( \sqrt{72} + \sqrt{50} \)?
2. How do you solve \( \log 10,000 \)?
3. Can you simplify \( \frac{2\sqrt{6}}{3\sqrt{2}} \)?
4. What is the value of \( 2 \log 100 \)?
5. How do you rationalize \( \frac{4}{\sqrt{3}} \)?

**Tip:** When working with logarithms, always try to express numbers as powers of the logarithm's base to simplify the calculation.

Simplify expressions with exponents, roots, and logarithms, including: 3 * 5^(5/2), 2*sqrt[3]{5} + 3*sqrt[3]{5} - sqrt[3]{5}, and log base operations.

This exercise covers the simplification of expressions involving powers, roots, and logarithms. Learn how to handle expressions such as 3 * 5^(5/2) and log 125 using key mathematical formulas like laws of exponents and logarithmic properties.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation