Questions: Rationalize the denominator of the expression. [ fracsqrta+sqrtbsqrta-sqrtb ]

$Rationalize the denominator of the expression. [ fracsqrta+sqrtbsqrta-sqrtb ]$

Transcript text: Rationalize the denominator of the expression. \[ \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} \]

Solution

Solution Steps

Step 1: Identify the Expression

We start with the expression: \[ \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} \]

Step 2: Multiply by the Conjugate

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{a} + \sqrt{b}$: \[ \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}} \cdot \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}} \]

Step 3: Simplify the Expression

After performing the multiplication, we have: \[ \frac{(\sqrt{a}+\sqrt{b})(\sqrt{a}+\sqrt{b})}{(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})} \]

Step 4: Factor the Numerator and Denominator

The numerator simplifies to: \[ (\sqrt{a}+\sqrt{b})^2 = a + 2\sqrt{ab} + b \] The denominator simplifies using the difference of squares: \[ (\sqrt{a})^2 - (\sqrt{b})^2 = a - b \]

Step 5: Write the Final Expression

Thus, the rationalized expression is: \[ \frac{a + 2\sqrt{ab} + b}{a - b} \]