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

8\( \sqrt{7} \) x 3\( \sqrt{4} \)
(8 x 3)\( \sqrt{7 \times 4} \)
24\( \sqrt{28} \)

Now we need to simplify the radical:

24\( \sqrt{28} \)
24\( \sqrt{7 \times 4} \)
24\( \sqrt{7 \times 2^2} \)
(24)(2)\( \sqrt{7} \)
48\( \sqrt{7} \)

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

4\( \sqrt{45} \) + 2\( \sqrt{5} \)
4\( \sqrt{9 \times 5} \) + 2\( \sqrt{5} \)
4\( \sqrt{3^2 \times 5} \) + 2\( \sqrt{5} \)
(4)(3)\( \sqrt{5} \) + 2\( \sqrt{5} \)
12\( \sqrt{5} \) + 2\( \sqrt{5} \)

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

To simplify a radical, factor out the perfect squares:

\( \sqrt{27} \)
\( \sqrt{9 \times 3} \)
\( \sqrt{3^2 \times 3} \)
3\( \sqrt{3} \)

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

\( \frac{12\sqrt{49}}{6\sqrt{7}} \)
\( \frac{12}{6} \) \( \sqrt{\frac{49}{7}} \)
2 \( \sqrt{7} \)

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

3\( \sqrt{50} \) - 2\( \sqrt{2} \)
3\( \sqrt{25 \times 2} \) - 2\( \sqrt{2} \)
3\( \sqrt{5^2 \times 2} \) - 2\( \sqrt{2} \)
(3)(5)\( \sqrt{2} \) - 2\( \sqrt{2} \)
15\( \sqrt{2} \) - 2\( \sqrt{2} \)

Now that the radicands are identical, you can subtract them: