Format the following question or task so that it is easier to read.
Fix any grammar or spelling errors.
Remove phrases that are not part of the question.
Do not remove or change LaTeX formatting.
Do not change or remove [tex] [/tex] tags.
If the question is nonsense, rewrite it so that it makes sense.
-----
Simplify the expression:
[tex] \frac{\sqrt{3}}{3-2 \sqrt{x}} [/tex]
-----

Response:



Answer :

To simplify the expression [tex]\(\frac{\sqrt{3}}{3 - 2 \sqrt{x}}\)[/tex], follow these steps:

1. Identify the Conjugate:
The given expression has a radical term in the denominator. To simplify, we'll multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 - 2 \sqrt{x}\)[/tex] is [tex]\(3 + 2 \sqrt{x}\)[/tex].

2. Multiply the Numerator and Denominator by the Conjugate:
[tex]\[ \frac{\sqrt{3}}{3 - 2 \sqrt{x}} \times \frac{3 + 2 \sqrt{x}}{3 + 2 \sqrt{x}} = \frac{\sqrt{3} (3 + 2 \sqrt{x})}{(3 - 2 \sqrt{x})(3 + 2 \sqrt{x})} \][/tex]

3. Simplify the Denominator:
The denominator is a difference of squares:
[tex]\[ (3 - 2 \sqrt{x})(3 + 2 \sqrt{x}) = 3^2 - (2 \sqrt{x})^2 = 9 - 4x \][/tex]

4. Expand the Numerator:
Distribute [tex]\(\sqrt{3}\)[/tex] across the terms in the numerator:
[tex]\[ \sqrt{3} (3 + 2 \sqrt{x}) = 3 \sqrt{3} + 2 \sqrt{3} \sqrt{x} \][/tex]

However, we can represent the original denominator [tex]\(3 - 2 \sqrt{x}\)[/tex] in a negative form:
[tex]\[ 3 - 2 \sqrt{x} = -(2 \sqrt{x} - 3) \][/tex]

Thus, rewriting the expression:
[tex]\[ \frac{\sqrt{3}}{3 - 2 \sqrt{x}} = \frac{\sqrt{3}}{-(2 \sqrt{x} - 3)} = -\frac{\sqrt{3}}{2 \sqrt{x} - 3} \][/tex]

Thus, the simplified form of the given expression is:
[tex]\[ -\frac{\sqrt{3}}{2 \sqrt{x} - 3} \][/tex]