$$ \frac{\sqrt3+\sqrt4}{\sqrt3-\sqrt4} $$

The given expression is:

\[
\frac{\sqrt{3} + \sqrt{4}}{\sqrt{3} - \sqrt{4}}
\]

Let's simplify it step by step.

### Step 1: Recognize that $\sqrt{4} = 2$.
So the expression becomes:

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

### Step 2: Multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of $\sqrt{3} - 2$ is $\sqrt{3} + 2$.

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

### Step 3: Simplify the denominator.
\[
(\sqrt{3})^2 - (2)^2 = 3 - 4 = -1
\]

### Step 4: Expand the numerator.
\[
(\sqrt{3} + 2)^2 = (\sqrt{3})^2 + 2 \times \sqrt{3} \times 2 + 2^2 = 3 + 4\sqrt{3} + 4 = 7 + 4\sqrt{3}
\]

### Step 5: Put it all together.
Now the expression becomes:

\[
\frac{7 + 4\sqrt{3}}{-1} = -(7 + 4\sqrt{3}) = -7 - 4\sqrt{3}
\]

Thus, the simplified expression is:

\[
\boxed{-7 - 4\sqrt{3}}
\]

Would you like further details on any step? Here are 5 related questions you might explore:

1. What is the importance of using the conjugate when simplifying such expressions?
2. How do you expand binomials like $(\sqrt{3} + 2)^2$?
3. Can you generalize this process for similar expressions with square roots?
4. What happens if the denominator does not have a difference of squares?
5. How does rationalizing affect the overall value of a fraction?

**Tip**: Multiplying by the conjugate is a powerful tool for simplifying expressions with radicals in the denominator!

Learn how to simplify the radical expression $$ \frac{\sqrt{3} + 2}{\sqrt{3} - 2} $$ using conjugates and binomial expansion. This step-by-step solution covers key algebraic concepts like radicals, conjugates, and the difference of squares. Ideal for students in Grades 9-11.

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