Questions: Rationalize the denominator: sqrt(3) / (7 - sqrt(2)) =

Transcript text: Rationalize the denominator: \[ \frac{\sqrt{3}}{7-\sqrt{2}}= \]

Solution

Solution Steps

To rationalize the denominator of the given fraction, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(7 - \sqrt{2}\) is \(7 + \sqrt{2}\). This will eliminate the square root in the denominator.

Step 1: Rationalizing the Denominator

To rationalize the denominator of the expression \( \frac{\sqrt{3}}{7 - \sqrt{2}} \), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 7 + \sqrt{2} \). This gives us:

\[ \frac{\sqrt{3}(7 + \sqrt{2})}{(7 - \sqrt{2})(7 + \sqrt{2})} \]

Step 2: Simplifying the Denominator

Next, we simplify the denominator using the difference of squares formula:

\[ (7 - \sqrt{2})(7 + \sqrt{2}) = 7^2 - (\sqrt{2})^2 = 49 - 2 = 47 \]

Step 3: Simplifying the Numerator

Now, we simplify the numerator:

\[ \sqrt{3}(7 + \sqrt{2}) = 7\sqrt{3} + \sqrt{6} \]

Step 4: Combining the Results

Putting it all together, we have:

\[ \frac{7\sqrt{3} + \sqrt{6}}{47} \]