We are tasked with rationalizing the expression:

\[
\frac{\sqrt{8}}{4 - \sqrt{6}}
\]

### Step 1: Simplify \(\sqrt{8}\)
First, simplify \(\sqrt{8}\):

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

Thus, the expression becomes:

\[
\frac{2\sqrt{2}}{4 - \sqrt{6}}
\]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of \(4 - \sqrt{6}\), which is \(4 + \sqrt{6}\). The conjugate helps eliminate the square root in the denominator.

So, multiply by:

\[
\frac{4 + \sqrt{6}}{4 + \sqrt{6}}
\]

This gives:

\[
\frac{2\sqrt{2}(4 + \sqrt{6})}{(4 - \sqrt{6})(4 + \sqrt{6})}
\]

### Step 3: Apply the difference of squares formula
In the denominator, use the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\), where \(a = 4\) and \(b = \sqrt{6}\):

\[
(4 - \sqrt{6})(4 + \sqrt{6}) = 4^2 - (\sqrt{6})^2 = 16 - 6 = 10
\]

Thus, the expression becomes:

\[
\frac{2\sqrt{2}(4 + \sqrt{6})}{10}
\]

### Step 4: Simplify the numerator
Now distribute \(2\sqrt{2}\) across the terms in the numerator:

\[
2\sqrt{2}(4 + \sqrt{6}) = 2\sqrt{2} \times 4 + 2\sqrt{2} \times \sqrt{6} = 8\sqrt{2} + 2\sqrt{12}
\]

Simplify \(\sqrt{12}\):

\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]

Thus, the numerator becomes:

\[
8\sqrt{2} + 2(2\sqrt{3}) = 8\sqrt{2} + 4\sqrt{3}
\]

### Step 5: Final expression
Now, the rationalized expression is:

\[
\frac{8\sqrt{2} + 4\sqrt{3}}{10}
\]

Simplify by dividing both terms in the numerator by 10:

\[
\frac{8\sqrt{2}}{10} + \frac{4\sqrt{3}}{10} = \frac{4\sqrt{2}}{5} + \frac{2\sqrt{3}}{5}
\]

Thus, the rationalized form of the expression is:

\[
\frac{4\sqrt{2}}{5} + \frac{2\sqrt{3}}{5}
\]

Would you like any more details or have any questions?

### Related Questions:
1. How do you rationalize expressions with cube roots?
2. What is the general method for simplifying radicals?
3. Can you explain the difference of squares formula in more detail?
4. How do you simplify complex fractions involving radicals?
5. What is the importance of rationalizing denominators?

### Tip:
Always look for conjugates when rationalizing denominators with square roots to simplify the process.

Learn how to rationalize the radical expression √8/(4 - √6) using conjugates and the difference of squares formula. This step-by-step guide simplifies the expression and is suitable for students in Grades 10-12 studying algebra and radical expressions.

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