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Resuelveme este problema solo quiero el proceso y nada de palabras solo el proceso

The expression to rationalize is:

\[
\frac{1}{\sqrt{3}+2} - \frac{4\sqrt{3}}{5-\sqrt{3}}
\]

Here is the step-by-step process to rationalize this expression:

### Step 1: Rationalize the first fraction
\[
\frac{1}{\sqrt{3} + 2}
\]
Multiply by the conjugate:
\[
\frac{1}{\sqrt{3}+2} \cdot \frac{\sqrt{3}-2}{\sqrt{3}-2} = \frac{\sqrt{3} - 2}{(\sqrt{3})^2 - (2)^2} = \frac{\sqrt{3} - 2}{3 - 4} = \frac{\sqrt{3} - 2}{-1} = -(\sqrt{3} - 2) = -\sqrt{3} + 2
\]

### Step 2: Rationalize the second fraction
\[
\frac{4\sqrt{3}}{5 - \sqrt{3}}
\]
Multiply by the conjugate:
\[
\frac{4\sqrt{3}}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}} = \frac{4\sqrt{3}(5 + \sqrt{3})}{(5)^2 - (\sqrt{3})^2} = \frac{4\sqrt{3}(5 + \sqrt{3})}{25 - 3} = \frac{4\sqrt{3}(5 + \sqrt{3})}{22}
\]

Now distribute:
\[
\frac{4\sqrt{3}(5 + \sqrt{3})}{22} = \frac{20\sqrt{3} + 12}{22} = \frac{10\sqrt{3} + 6}{11}
\]

### Step 3: Combine the results
Now, subtract the two rationalized expressions:
\[
-\sqrt{3} + 2 - \frac{10\sqrt{3} + 6}{11}
\]

To subtract these, convert the first term to have a common denominator:
\[
-\sqrt{3} + 2 = \frac{-11\sqrt{3} + 22}{11}
\]

So now we subtract:
\[
\frac{-11\sqrt{3} + 22}{11} - \frac{10\sqrt{3} + 6}{11} = \frac{-11\sqrt{3} + 22 - 10\sqrt{3} - 6}{11} = \frac{-21\sqrt{3} + 16}{11}
\]

Final answer:
\[
\frac{-21\sqrt{3} + 16}{11}
\]

Let me know if you need further clarifications!

Racionalizar la siguiente expresión: \frac{1}{\sqrt{3}+2} - \frac{4\sqrt{3}}{5-\sqrt{3}}

Learn how to rationalize the expression 1/(√3 + 2) - 4√3/(5 - √3) using step-by-step instructions. This problem covers key algebraic concepts, including rationalization by multiplying by the conjugate and properties of radicals. Suitable for high school students in Grades 9-12.

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