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

6\( \sqrt{75} \) + 3\( \sqrt{3} \)
6\( \sqrt{25 \times 3} \) + 3\( \sqrt{3} \)
6\( \sqrt{5^2 \times 3} \) + 3\( \sqrt{3} \)
(6)(5)\( \sqrt{3} \) + 3\( \sqrt{3} \)
30\( \sqrt{3} \) + 3\( \sqrt{3} \)

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

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

\( \frac{12\sqrt{10}}{6\sqrt{5}} \)
\( \frac{12}{6} \) \( \sqrt{\frac{10}{5}} \)
2 \( \sqrt{2} \)

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

3\( \sqrt{20} \) - 7\( \sqrt{5} \)
3\( \sqrt{4 \times 5} \) - 7\( \sqrt{5} \)
3\( \sqrt{2^2 \times 5} \) - 7\( \sqrt{5} \)
(3)(2)\( \sqrt{5} \) - 7\( \sqrt{5} \)
6\( \sqrt{5} \) - 7\( \sqrt{5} \)

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

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

7\( \sqrt{5} \) x 4\( \sqrt{5} \)
(7 x 4)\( \sqrt{5 \times 5} \)
28\( \sqrt{25} \)

Now we need to simplify the radical:

28\( \sqrt{25} \)
28\( \sqrt{5^2} \)
(28)(5)
140

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:

\( \sqrt{\frac{64}{49}} \)
\( \frac{\sqrt{64}}{\sqrt{49}} \)
\( \frac{\sqrt{8^2}}{\sqrt{7^2}} \)
\( \frac{8}{7} \)
1\(\frac{1}{7}\)