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

8\( \sqrt{50} \) + 7\( \sqrt{2} \)
8\( \sqrt{25 \times 2} \) + 7\( \sqrt{2} \)
8\( \sqrt{5^2 \times 2} \) + 7\( \sqrt{2} \)
(8)(5)\( \sqrt{2} \) + 7\( \sqrt{2} \)
40\( \sqrt{2} \) + 7\( \sqrt{2} \)

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

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:

\( \sqrt{\frac{49}{81}} \)
\( \frac{\sqrt{49}}{\sqrt{81}} \)
\( \frac{\sqrt{7^2}}{\sqrt{9^2}} \)
\(\frac{7}{9}\)

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

5\( \sqrt{8} \) x 3\( \sqrt{6} \)
(5 x 3)\( \sqrt{8 \times 6} \)
15\( \sqrt{48} \)

Now we need to simplify the radical:

15\( \sqrt{48} \)
15\( \sqrt{3 \times 16} \)
15\( \sqrt{3 \times 4^2} \)
(15)(4)\( \sqrt{3} \)
60\( \sqrt{3} \)

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

\( \frac{8\sqrt{35}}{4\sqrt{7}} \)
\( \frac{8}{4} \) \( \sqrt{\frac{35}{7}} \)
2 \( \sqrt{5} \)

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

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

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