https://file.aiquickdraw.com/mathbot/file/1725586274330ock7l40z.jpg

The expression shown in the image is:

\[
\frac{6}{4 + \sqrt{15}}
\]

To simplify this, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4 + \sqrt{15}\) is \(4 - \sqrt{15}\).

Let's perform the rationalization:

\[
\frac{6}{4 + \sqrt{15}} \times \frac{4 - \sqrt{15}}{4 - \sqrt{15}} = \frac{6(4 - \sqrt{15})}{(4 + \sqrt{15})(4 - \sqrt{15})}
\]

Now, I'll calculate the result.The simplified form of the expression is:

\[
\frac{6}{4 + \sqrt{15}} = 24 - 6\sqrt{15}
\]

This is the result after rationalizing the denominator.

Would you like more details or have any questions?

Here are five related questions you might want to explore:
1. How can you rationalize other similar expressions with square roots in the denominator?
2. What is the difference between conjugates and how are they used in simplifications?
3. Can this method be extended to more complex denominators involving multiple radicals?
4. How does the result change if the numerator was different, such as \( \frac{8}{4 + \sqrt{15}} \)?
5. What is the geometrical interpretation of rationalizing a denominator?

**Tip**: Rationalizing the denominator helps avoid irrational numbers in the denominator, making expressions easier to handle, especially in further calculations.

Simplify the expression 6 / (4 + sqrt(15)).

Learn how to simplify the expression 6 / (4 + sqrt(15)) by rationalizing the denominator. This process involves using the conjugate and applying the difference of squares formula. Ideal for high school students learning algebra and square roots.

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