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:

\( \sqrt{\frac{25}{64}} \)
\( \frac{\sqrt{25}}{\sqrt{64}} \)
\( \frac{\sqrt{5^2}}{\sqrt{8^2}} \)
\(\frac{5}{8}\)

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

9\( \sqrt{12} \) - 6\( \sqrt{3} \)
9\( \sqrt{4 \times 3} \) - 6\( \sqrt{3} \)
9\( \sqrt{2^2 \times 3} \) - 6\( \sqrt{3} \)
(9)(2)\( \sqrt{3} \) - 6\( \sqrt{3} \)
18\( \sqrt{3} \) - 6\( \sqrt{3} \)

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

To simplify a radical, factor out the perfect squares:

\( \sqrt{112} \)
\( \sqrt{16 \times 7} \)
\( \sqrt{4^2 \times 7} \)
4\( \sqrt{7} \)

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

\( \frac{8\sqrt{12}}{4\sqrt{3}} \)
\( \frac{8}{4} \) \( \sqrt{\frac{12}{3}} \)
2 \( \sqrt{4} \)

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

5\( \sqrt{18} \) + 9\( \sqrt{2} \)
5\( \sqrt{9 \times 2} \) + 9\( \sqrt{2} \)
5\( \sqrt{3^2 \times 2} \) + 9\( \sqrt{2} \)
(5)(3)\( \sqrt{2} \) + 9\( \sqrt{2} \)
15\( \sqrt{2} \) + 9\( \sqrt{2} \)

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