https://file.aiquickdraw.com/mathbot/file/17265319731349lxkhemm.jpg

The image asks to rationalize the expression:

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

To rationalize the denominator, we will multiply both the numerator and denominator by the conjugate of the denominator, which is \(2\sqrt{3} + 3\sqrt{5}\). Let's walk through the steps.

The expression becomes:

\[
\frac{\sqrt{3} + \sqrt{5}}{2\sqrt{3} - 3\sqrt{5}} \times \frac{2\sqrt{3} + 3\sqrt{5}}{2\sqrt{3} + 3\sqrt{5}}
\]

Now let's expand both the numerator and the denominator.

1. **Numerator**:
   \[
   (\sqrt{3} + \sqrt{5})(2\sqrt{3} + 3\sqrt{5}) = \sqrt{3} \cdot 2\sqrt{3} + \sqrt{3} \cdot 3\sqrt{5} + \sqrt{5} \cdot 2\sqrt{3} + \sqrt{5} \cdot 3\sqrt{5}
   \]

   Simplifying:
   \[
   = 2\cdot3 + 3\sqrt{15} + 2\sqrt{15} + 3\cdot5 = 6 + 5\sqrt{15} + 15 = 21 + 5\sqrt{15}
   \]

2. **Denominator**:
   This is a difference of squares:
   \[
   (2\sqrt{3} - 3\sqrt{5})(2\sqrt{3} + 3\sqrt{5}) = (2\sqrt{3})^2 - (3\sqrt{5})^2 = 4 \cdot 3 - 9 \cdot 5 = 12 - 45 = -33
   \]

So the rationalized form is:

\[
\frac{21 + 5\sqrt{15}}{-33}
\]

or, simplified:

\[
-\frac{21 + 5\sqrt{15}}{33}
\]

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How can you simplify expressions with radicals further?
2. What is the conjugate, and why is it useful in rationalizing denominators?
3. Can all expressions with square roots be rationalized?
4. How do you handle cases where both the numerator and denominator contain complex radicals?
5. What are some common mistakes when rationalizing denominators?

**Tip**: Always ensure to multiply both the numerator and the denominator by the conjugate when dealing with irrational numbers in the denominator.

Bentuk rasional dari (sqrt(3) + sqrt(5)) / (2sqrt(3) - 3sqrt(5)) adalah?

Solve the expression (sqrt(3) + sqrt(5)) / (2sqrt(3) - 3sqrt(5)) by rationalizing the denominator using the conjugate method. This algebraic problem helps students understand how to handle radicals and difference of squares.

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