Which of the following is equivalent to `(sqrt(a)timessqrt(b))/(3sqrt(a)-2sqrt(b))`?

Options

A. `(sqrt(b))/(sqrt(a))`
B. `sqrt(b)`
C. `(2sqrt(a))/(b)`
D. `sqrt(ab)`

Answer

To solve the expression \(\frac{\sqrt{a} \times \sqrt{b}}{3\sqrt{a} - 2\sqrt{b}}\), we can follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ \frac{\sqrt{a} \times \sqrt{b}}{3\sqrt{a} - 2\sqrt{b}} \]

### Step 2: Factor out \(\sqrt{a}\) from the denominator
In the denominator, we can factor out \(\sqrt{a}\):
\[ 3\sqrt{a} - 2\sqrt{b} = \sqrt{a}(3 - 2\frac{\sqrt{b}}{\sqrt{a}}) \]

### Step 3: Substitute the factored form back into the expression
Now, substituting the factored form back into the expression gives us:
\[ \frac{\sqrt{a} \times \sqrt{b}}{\sqrt{a}(3 - 2\frac{\sqrt{b}}{\sqrt{a}})} \]

### Step 4: Cancel \(\sqrt{a}\) from the numerator and denominator
Since \(\sqrt{a}\) is common in both the numerator and the denominator, we can cancel it out (assuming \(\sqrt{a} \neq 0\)):
\[ \frac{\sqrt{b}}{3 - 2\frac{\sqrt{b}}{\sqrt{a}}} \]

### Step 5: Simplify further if possible
However, we can also express the denominator in a simpler form:
\[ 3 - 2\frac{\sqrt{b}}{\sqrt{a}} = 3 - 2\sqrt{\frac{b}{a}} \]
Thus, the expression simplifies to:
\[ \frac{\sqrt{b}}{3 - 2\sqrt{\frac{b}{a}}} \]

### Final Result
The expression \(\frac{\sqrt{a} \times \sqrt{b}}{3\sqrt{a} - 2\sqrt{b}}\) is equivalent to:
\[ \frac{\sqrt{b}}{3 - 2\sqrt{\frac{b}{a}}} \]