Question

use the work shown below to answer the following question.

\\\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}\left(\frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}}\
ight)\\
\\=\frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x})}\\
\\=\frac{\sqrt{3}(\sqrt{3})+\sqrt{3}(\sqrt{x})}{(\sqrt{3})^2-(\sqrt{x})^2}\\

what is \\(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}\\) in simplest form?

\\(\circ\\) \\(\frac{9+\sqrt{3x}}{3-x}\\)

\\(\circ\\) \\(\frac{3+\sqrt{3x}}{3-x}\\)

\\(\circ\\) \\(\frac{9+\sqrt{3x}}{3+x}\\)

\\(\circ\\) \\(\frac{\sqrt{3}+\sqrt{x}}{3+x}\\)

Explanation:

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<reasoning>

Multiply by the conjugate

Using the Rationalizing the Denominator knowledge point
\[
\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} \cdot \frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}} = \frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x})}
\]

Simplify the numerator

Using the Product Property of Radicals and Algebraic Simplification knowledge points
\[
\sqrt{3}(\sqrt{3}) + \sqrt{3}(\sqrt{x}) = 3 + \sqrt{3x}
\]

Simplify the denominator

Using the Algebraic Simplification knowledge point
\[
(\sqrt{3})^2 - (\sqrt{x})^2 = 3 - x
\]

Combine the simplified parts

Using the Algebraic Simplification knowledge point
\[
\frac{3 + \sqrt{3x}}{3 - x}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{9+\sqrt{3x}}{3-x}\)</mcq-option>
<mcq-correct>(B) \(\frac{3+\sqrt{3x}}{3-x}\)</mcq-correct>
<mcq-option>(C) \(\frac{9+\sqrt{3x}}{3+x}\)</mcq-option>
<mcq-option>(D) \(\frac{\sqrt{3}+\sqrt{x}}{3+x}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Multiply by the conjugate

Using the Rationalizing the Denominator knowledge point
\[
\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} \cdot \frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}} = \frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x})}
\]

Simplify the numerator

Using the Product Property of Radicals and Algebraic Simplification knowledge points
\[
\sqrt{3}(\sqrt{3}) + \sqrt{3}(\sqrt{x}) = 3 + \sqrt{3x}
\]

Simplify the denominator

Using the Algebraic Simplification knowledge point
\[
(\sqrt{3})^2 - (\sqrt{x})^2 = 3 - x
\]

Combine the simplified parts

Using the Algebraic Simplification knowledge point
\[
\frac{3 + \sqrt{3x}}{3 - x}
\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{9+\sqrt{3x}}{3-x}\)</mcq-option>
<mcq-correct>(B) \(\frac{3+\sqrt{3x}}{3-x}\)</mcq-correct>
<mcq-option>(C) \(\frac{9+\sqrt{3x}}{3+x}\)</mcq-option>
<mcq-option>(D) \(\frac{\sqrt{3}+\sqrt{x}}{3+x}\)</mcq-option>
</answer>

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