Express as a single fraction in simplest radical form with a rational denominator.

start fraction, square root of, 3, end square root, minus, square root of, 8, end square root, divided by, square root of, 8, end square root, plus, square root of, 12, end square root, end fraction

8​+12​3​−8​​


We need to simplify the following expression:

\[
\frac{\sqrt{3} - \sqrt{8}}{\sqrt{8} + \sqrt{12}}
\]

### Step 1: Simplify the square roots.

- \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\)
- \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)

Substitute these into the expression:

\[
\frac{\sqrt{3} - 2\sqrt{2}}{2\sqrt{2} + 2\sqrt{3}}
\]

### Step 2: Rationalize the denominator.

To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(2\sqrt{2} - 2\sqrt{3}\):

\[
\frac{(\sqrt{3} - 2\sqrt{2})(2\sqrt{2} - 2\sqrt{3})}{(2\sqrt{2} + 2\sqrt{3})(2\sqrt{2} - 2\sqrt{3})}
\]

### Step 3: Expand the numerator and denominator.

#### Denominator:
Use the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\).

\[
(2\sqrt{2})^2 - (2\sqrt{3})^2 = 4 \times 2 - 4 \times 3 = 8 - 12 = -4
\]

#### Numerator:
Now, expand \((\sqrt{3} - 2\sqrt{2})(2\sqrt{2} - 2\sqrt{3})\) using the distributive property:

\[
= \sqrt{3}(2\sqrt{2}) - \sqrt{3}(2\sqrt{3}) - 2\sqrt{2}(2\sqrt{2}) + 2\sqrt{2}(2\sqrt{3})
\]
\[
= 2\sqrt{6} - 6 - 4 + 4\sqrt{6}
\]
Combine like terms:

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

### Step 4: Put it all together.

Now, substitute the expanded numerator and denominator back into the expression:

\[
\frac{6\sqrt{6} - 10}{-4}
\]

### Step 5: Simplify the result.

Divide each term by \(-4\):

\[
\frac{6\sqrt{6}}{-4} - \frac{10}{-4} = -\frac{3\sqrt{6}}{2} + \frac{5}{2}
\]

### Final Answer:
\[
\boxed{\frac{5}{2} - \frac{3\sqrt{6}}{2}}
\]

Would you like further clarification or more details on any step? 

Here are 5 related questions for practice:
1. Simplify \(\frac{3 + \sqrt{2}}{2 - \sqrt{2}}\).
2. Simplify \(\frac{5 - \sqrt{3}}{2 + \sqrt{3}}\).
3. Rationalize the denominator of \(\frac{1}{\sqrt{6} - \sqrt{2}}\).
4. Simplify \(\frac{7 + 2\sqrt{5}}{1 - \sqrt{5}}\).
5. Simplify \(\frac{\sqrt{12} - \sqrt{18}}{\sqrt{6}}\).

**Tip:** When rationalizing, always multiply by the conjugate of the denominator to eliminate radicals.

Express as a single fraction in simplest radical form with a rational denominator. \(\frac{\sqrt{3} - \sqrt{8}}{\sqrt{8} + \sqrt{12}}\)

This problem focuses on simplifying the radical expression \(\frac{\sqrt{3} - \sqrt{8}}{\sqrt{8} + \sqrt{12}}\), involving the simplification of square roots and rationalization of the denominator using the difference of squares formula. Suitable for students in Grades 9-11.

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