To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{4}{16}} \)
\( \frac{\sqrt{4}}{\sqrt{16}} \)
\( \frac{\sqrt{2^2}}{\sqrt{4^2}} \)
\(\frac{1}{2}\)

To add these radicals together their radicands must be the same:

4\( \sqrt{48} \) + 8\( \sqrt{3} \)
4\( \sqrt{16 \times 3} \) + 8\( \sqrt{3} \)
4\( \sqrt{4^2 \times 3} \) + 8\( \sqrt{3} \)
(4)(4)\( \sqrt{3} \) + 8\( \sqrt{3} \)
16\( \sqrt{3} \) + 8\( \sqrt{3} \)

Now that the radicands are identical, you can add them together:

To multiply terms with radicals, multiply the coefficients and radicands separately:

3\( \sqrt{3} \) x 7\( \sqrt{6} \)
(3 x 7)\( \sqrt{3 \times 6} \)
21\( \sqrt{18} \)

Now we need to simplify the radical:

21\( \sqrt{18} \)
21\( \sqrt{2 \times 9} \)
21\( \sqrt{2 \times 3^2} \)
(21)(3)\( \sqrt{2} \)
63\( \sqrt{2} \)

To divide terms with radicals, divide the coefficients and radicands separately:

\( \frac{42\sqrt{14}}{6\sqrt{2}} \)
\( \frac{42}{6} \) \( \sqrt{\frac{14}{2}} \)
7 \( \sqrt{7} \)

To simplify a radical, factor out the perfect squares:

\( \sqrt{18} \)
\( \sqrt{9 \times 2} \)
\( \sqrt{3^2 \times 2} \)
3\( \sqrt{2} \)