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

9\( \sqrt{4} \) x 2\( \sqrt{8} \)
(9 x 2)\( \sqrt{4 \times 8} \)
18\( \sqrt{32} \)

Now we need to simplify the radical:

18\( \sqrt{32} \)
18\( \sqrt{2 \times 16} \)
18\( \sqrt{2 \times 4^2} \)
(18)(4)\( \sqrt{2} \)
72\( \sqrt{2} \)

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:

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

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

4\( \sqrt{125} \) - 9\( \sqrt{5} \)
4\( \sqrt{25 \times 5} \) - 9\( \sqrt{5} \)
4\( \sqrt{5^2 \times 5} \) - 9\( \sqrt{5} \)
(4)(5)\( \sqrt{5} \) - 9\( \sqrt{5} \)
20\( \sqrt{5} \) - 9\( \sqrt{5} \)

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

To simplify a radical, factor out the perfect squares:

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