What is the rationalized form of sqrt(3)/(2sqrt(7) - 4)?

To find the rationalized form of (\frac{\sqrt{3}}{2\sqrt{7} - 4}), we can eliminate the radical in the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (2\sqrt{7} + 4).

Starting with the expression:

[

\frac{\sqrt{3}}{2\sqrt{7} - 4} \cdot \frac{2\sqrt{7} + 4}{2\sqrt{7} + 4} = \frac{\sqrt{3}(2\sqrt{7} + 4)}{(2\sqrt{7} - 4)(2\sqrt{7} + 4)}

]

The denominator simplifies using the difference of squares:

[

(2\sqrt{7})^2 - 4^2 = 28 - 16 = 12

]

Now, focusing on the numerator:

[

\sqrt{3}(2\sqrt{7}) + \sqrt{3}(4) = 2\sqrt{21} + 4\sqrt{3}

]

Thus, our expression now looks like:

[

\