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

9\( \sqrt{63} \) - 8\( \sqrt{7} \)
9\( \sqrt{9 \times 7} \) - 8\( \sqrt{7} \)
9\( \sqrt{3^2 \times 7} \) - 8\( \sqrt{7} \)
(9)(3)\( \sqrt{7} \) - 8\( \sqrt{7} \)
27\( \sqrt{7} \) - 8\( \sqrt{7} \)

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

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

7\( \sqrt{3} \) x 7\( \sqrt{7} \)
(7 x 7)\( \sqrt{3 \times 7} \)
49\( \sqrt{21} \)

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

9\( \sqrt{125} \) + 7\( \sqrt{5} \)
9\( \sqrt{25 \times 5} \) + 7\( \sqrt{5} \)
9\( \sqrt{5^2 \times 5} \) + 7\( \sqrt{5} \)
(9)(5)\( \sqrt{5} \) + 7\( \sqrt{5} \)
45\( \sqrt{5} \) + 7\( \sqrt{5} \)

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

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

\( \frac{14\sqrt{20}}{7\sqrt{4}} \)
\( \frac{14}{7} \) \( \sqrt{\frac{20}{4}} \)
2 \( \sqrt{5} \)

To simplify a radical, factor out the perfect squares:

\( \sqrt{175} \)
\( \sqrt{25 \times 7} \)
\( \sqrt{5^2 \times 7} \)
5\( \sqrt{7} \)