20. $$ \frac{\sqrt{6}}{\sqrt{2}+\sqrt{3}} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$ \frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} $$

we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of $$\sqrt{2} + \sqrt{3}$$ is $$\sqrt{2} - \sqrt{3}$$.

**Step 1: Multiply by the Conjugate**

$$
\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{\sqrt{6}(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})}
$$

**Step 2: Simplify the Denominator**

$$
(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1
$$

**Step 3: Distribute in the Numerator**

$$
\sqrt{6}(\sqrt{2} - \sqrt{3}) = \sqrt{6} \cdot \sqrt{2} - \sqrt{6} \cdot \sqrt{3} = \sqrt{12} - \sqrt{18}
$$

Simplify the square roots:

$$
\sqrt{12} = 2\sqrt{3} \quad \text{and} \quad \sqrt{18} = 3\sqrt{2}
$$

So,

$$
\sqrt{12} - \sqrt{18} = 2\sqrt{3} - 3\sqrt{2}
$$

**Step 4: Combine Everything**

$$
\frac{2\sqrt{3} - 3\sqrt{2}}{-1} = -2\sqrt{3} + 3\sqrt{2} = 3\sqrt{2} - 2\sqrt{3}
$$

**Final Answer:**

$$
\boxed{3\sqrt{2} - 2\sqrt{3}}
$$
The simplified form of the expression is $$ 3\sqrt{2} - 2\sqrt{3} $$.

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