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

\( \frac{20\sqrt{40}}{4\sqrt{8}} \)
\( \frac{20}{4} \) \( \sqrt{\frac{40}{8}} \)
5 \( \sqrt{5} \)

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

4\( \sqrt{175} \) + 4\( \sqrt{7} \)
4\( \sqrt{25 \times 7} \) + 4\( \sqrt{7} \)
4\( \sqrt{5^2 \times 7} \) + 4\( \sqrt{7} \)
(4)(5)\( \sqrt{7} \) + 4\( \sqrt{7} \)
20\( \sqrt{7} \) + 4\( \sqrt{7} \)

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:

\( \sqrt{\frac{81}{81}} \)
\( \frac{\sqrt{81}}{\sqrt{81}} \)
\( \frac{\sqrt{9^2}}{\sqrt{9^2}} \)
1

To simplify a radical, factor out the perfect squares:

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

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

4\( \sqrt{9} \) x 9\( \sqrt{8} \)
(4 x 9)\( \sqrt{9 \times 8} \)
36\( \sqrt{72} \)

Now we need to simplify the radical:

36\( \sqrt{72} \)
36\( \sqrt{2 \times 36} \)
36\( \sqrt{2 \times 6^2} \)
(36)(6)\( \sqrt{2} \)
216\( \sqrt{2} \)