Rationalize the denominator and simplify. $$ \frac{2 \sqrt{x y}-3 \sqrt{y}}{\sqrt{x}}, x0, y \geq 0 $$ $$ -\sqrt{x y} $$ $$ \frac{2 x \sqrt{y}-3 \sqrt{x y}}{x} $$ $$ \frac{-\sqrt{x}}{x} $$ $$ \frac{2 x-\sqrt{y}}{x} $$

Question

Rationalize the denominator and simplify. $$ \frac{2 \sqrt{x y}-3 \sqrt{y}}{\sqrt{x}}, x>0, y \geq 0 $$ $$ -\sqrt{x y} $$ $$ \frac{2 x \sqrt{y}-3 \sqrt{x y}}{x} $$ $$ \frac{-\sqrt{x}}{x} $$ $$ \frac{2 x-\sqrt{y}}{x} $$

UpStudy AI · Accepted Answer

To rationalize the denominator and simplify the expression:

$$
\frac{2 \sqrt{x y} - 3 \sqrt{y}}{\sqrt{x}}, \quad x > 0, \ y \geq 0
$$

**Step 1: Rationalize the Denominator**

Multiply the numerator and the denominator by $$\sqrt{x}$$ to eliminate the square root from the denominator:

$$
\frac{2 \sqrt{x y} - 3 \sqrt{y}}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{(2 \sqrt{x y} - 3 \sqrt{y}) \cdot \sqrt{x}}{x}
$$

**Step 2: Simplify the Numerator**

Distribute $$\sqrt{x}$$ in the numerator:

$$
(2 \sqrt{x y} - 3 \sqrt{y}) \cdot \sqrt{x} = 2 \sqrt{x} \cdot \sqrt{x y} - 3 \sqrt{y} \cdot \sqrt{x}
$$

Simplify each term:

$$
2 \sqrt{x} \cdot \sqrt{x y} = 2 x \sqrt{y}
$$
$$
3 \sqrt{y} \cdot \sqrt{x} = 3 \sqrt{x y}
$$

So the numerator becomes:

$$
2 x \sqrt{y} - 3 \sqrt{x y}
$$

**Step 3: Combine the Results**

Now, place the simplified numerator over the denominator:

$$
\frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x}
$$

This expression can also be factored if desired:

$$
\frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x} = \frac{2 x \sqrt{y}}{x} - \frac{3 \sqrt{x y}}{x} = 2 \sqrt{y} - \frac{3 \sqrt{x y}}{x}
$$

However, the fully rationalized and simplified form among the provided options is:

$$
\frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x}
$$

**Final Answer:**

$$
\frac{2 x \sqrt{y}-3 \sqrt{x y}}{x}
$$
$$ \frac{2 x \sqrt{y} - 3 \sqrt{x y}}{x} $$

Rationalize the denominator and simplify.

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