Answer
$7 + 4\sqrt {3}$
Work Step by Step
In this problem, the first thing we have to do is simplify the radicals by rewriting the radicands as the product of a perfect square and another number:
$\dfrac{4 + \sqrt {4 \cdot 3}}{4 - \sqrt {4 \cdot 3}}$
Now, simplify the radicands by taking the perfect squares:
$\dfrac{4 + 2\sqrt {3}}{4 - 2\sqrt {3}}$
We do not want to deal with radicals in the denominator, so we need to get rid of the radical in the denominator by multiplying both numerator and denominator by the conjugate of the denominator:
$\left(\dfrac{4 + 2\sqrt {3}}{4 - 2\sqrt {3}}\right)\left(\dfrac{4 + 2\sqrt {3}}{4 + 2\sqrt {3}}\right)$
Use the FOIL method to expand the numerator and denominator:
$\dfrac{(4)(4) + (4)(2\sqrt {3}) + (2\sqrt {3})(4) + (2\sqrt {3})(2\sqrt {3})}{(4)(4) + (4)(2\sqrt {3}) - (2\sqrt {3})(4) - (2\sqrt {3})(2\sqrt {3})}$
Multiply to simplify:
$\dfrac{(16) + (8\sqrt {3}) + (8\sqrt {3}) + (4\sqrt {3 • 3})}{(16) + (8\sqrt {3}) - (8\sqrt {3}) - (4\sqrt {3 • 3})}$
Simplify radicals:
$\dfrac{16 + 8\sqrt {3} + 8\sqrt {3} + 12}{16 + 8\sqrt {3} - 8\sqrt {3} - 12}$
Combine like terms:
$\dfrac{28 + 16\sqrt {3}}{4}$
Simplify the fraction by dividing both numerator and denominator by their greatest common factor, $4$:
$7 + 4\sqrt {3}$
