3√2÷ √6 + √3 + √6 ÷ √2 + √3 - 4√3 ÷ √6 + √2 = 0

Let's solve the equation step by step:

We are given:

\[
\frac{3\sqrt{2}}{\sqrt{6}} + \sqrt{3} + \frac{\sqrt{6}}{\sqrt{2}} + \sqrt{3} - \frac{4\sqrt{3}}{\sqrt{6}} + \sqrt{2} = 0
\]

### Step 1: Simplify each term involving square roots

We will simplify each fraction individually by rationalizing the denominators where needed:

1. **Simplify** \( \frac{3\sqrt{2}}{\sqrt{6}} \):
   \[
   \frac{3\sqrt{2}}{\sqrt{6}} = \frac{3\sqrt{2}}{\sqrt{2} \cdot \sqrt{3}} = \frac{3}{\sqrt{3}} = 3 \cdot \frac{\sqrt{3}}{3} = \sqrt{3}
   \]

2. **Simplify** \( \frac{\sqrt{6}}{\sqrt{2}} \):
   \[
   \frac{\sqrt{6}}{\sqrt{2}} = \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{2}} = \sqrt{3}
   \]

3. **Simplify** \( \frac{4\sqrt{3}}{\sqrt{6}} \):
   \[
   \frac{4\sqrt{3}}{\sqrt{6}} = \frac{4\sqrt{3}}{\sqrt{2} \cdot \sqrt{3}} = \frac{4}{\sqrt{2}} = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2}
   \]

### Step 2: Substitute these simplified expressions into the equation

Now we substitute the simplified values back into the equation:
\[
\sqrt{3} + \sqrt{3} + \sqrt{3} + \sqrt{2} - 2\sqrt{2} + \sqrt{2} = 0
\]

### Step 3: Combine like terms

- Combine the \( \sqrt{3} \) terms:
  \[
  \sqrt{3} + \sqrt{3} + \sqrt{3} = 3\sqrt{3}
  \]

- Combine the \( \sqrt{2} \) terms:
  \[
  \sqrt{2} - 2\sqrt{2} + \sqrt{2} = 0
  \]

### Step 4: Final equation

Now we are left with:
\[
3\sqrt{3} = 0
\]

This is not possible, because \( \sqrt{3} \) is a non-zero number. Hence, the equation has no solutions.

### Conclusion:
There are no real solutions to the given equation.

Would you like more details or have any further questions?

Here are 5 related questions for you:
1. How do you simplify expressions with square roots?
2. What does rationalizing a denominator mean in mathematics?
3. Can you have an equation equal to zero if one side has non-zero terms?
4. How are terms with square roots combined in algebraic expressions?
5. What are common techniques for solving radical equations?

**Tip:** Always simplify the square roots and combine like terms before solving an equation!

3√2 ÷ √6 + √3 + √6 ÷ √2 + √3 - 4√3 ÷ √6 + √2 = 0

Learn how to solve a complex radical equation step-by-step. This problem focuses on simplifying radical expressions and rationalizing denominators, suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation