Rationalize the denominator and simplify: (sqrt (b))/(3+sqrt (b)) (3sqrt (b)-b)/(9-b) (3-b)/(9-b) (2sqrt (b))/(9-b) 1 (1)/(3)

## Step 1: Identify the problem<br />### The goal is to rationalize the denominator of \( \frac{\sqrt{b}}{3+\sqrt{b}} \) and simplify it step-by-step.<br /><br />## Step 2: Multiply numerator and denominator by the conjugate<br />### To rationalize the denominator, multiply both numerator and denominator by the conjugate of \( 3+\sqrt{b} \), which is \( 3-\sqrt{b} \). This eliminates the square root in the denominator:<br />\[<br />\frac{\sqrt{b}}{3+\sqrt{b}} \cdot \frac{3-\sqrt{b}}{3-\sqrt{b}} = \frac{\sqrt{b}(3-\sqrt{b})}{(3+\sqrt{b})(3-\sqrt{b})}.<br />\]<br /><br />## Step 3: Simplify the denominator<br />### Use the difference of squares formula: \( (a+b)(a-b) = a^2 - b^2 \). Here, \( a = 3 \) and \( b = \sqrt{b} \):<br />\[<br />(3+\sqrt{b})(3-\sqrt{b}) = 3^2 - (\sqrt{b})^2 = 9 - b.<br />\]<br /><br />## Step 4: Expand the numerator<br />### Distribute \( \sqrt{b} \) across \( (3-\sqrt{b}) \):<br />\[<br />\sqrt{b}(3-\sqrt{b}) = 3\sqrt{b} - b.<br />\]<br /><br />## Step 5: Combine results<br />### Substitute the simplified numerator and denominator back into the fraction:<br />\[<br />\frac{\sqrt{b}(3-\sqrt{b})}{(3+\sqrt{b})(3-\sqrt{b})} = \frac{3\sqrt{b} - b}{9-b}.<br />\]<br /><br />## Step 6: Verify simplification options<br />### Among the given options, \( \frac{3\sqrt{b} - b}{9-b} \) matches the simplified form. No further simplifications are possible.