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

7\( \sqrt{3} \) x 7\( \sqrt{4} \)
(7 x 7)\( \sqrt{3 \times 4} \)
49\( \sqrt{12} \)

Now we need to simplify the radical:

49\( \sqrt{12} \)
49\( \sqrt{3 \times 4} \)
49\( \sqrt{3 \times 2^2} \)
(49)(2)\( \sqrt{3} \)
98\( \sqrt{3} \)

To simplify a radical, factor out the perfect squares:

\( \sqrt{112} \)
\( \sqrt{16 \times 7} \)
\( \sqrt{4^2 \times 7} \)
4\( \sqrt{7} \)

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

8\( \sqrt{75} \) - 2\( \sqrt{3} \)
8\( \sqrt{25 \times 3} \) - 2\( \sqrt{3} \)
8\( \sqrt{5^2 \times 3} \) - 2\( \sqrt{3} \)
(8)(5)\( \sqrt{3} \) - 2\( \sqrt{3} \)
40\( \sqrt{3} \) - 2\( \sqrt{3} \)

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

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

3\( \sqrt{28} \) + 7\( \sqrt{7} \)
3\( \sqrt{4 \times 7} \) + 7\( \sqrt{7} \)
3\( \sqrt{2^2 \times 7} \) + 7\( \sqrt{7} \)
(3)(2)\( \sqrt{7} \) + 7\( \sqrt{7} \)
6\( \sqrt{7} \) + 7\( \sqrt{7} \)

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

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

\( \frac{10\sqrt{20}}{2\sqrt{4}} \)
\( \frac{10}{2} \) \( \sqrt{\frac{20}{4}} \)
5 \( \sqrt{5} \)