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:

\( \sqrt{\frac{64}{81}} \)
\( \frac{\sqrt{64}}{\sqrt{81}} \)
\( \frac{\sqrt{8^2}}{\sqrt{9^2}} \)
\(\frac{8}{9}\)

You have 3 out of the total of 150 raffle tickets sold so you have a (\( \frac{3}{150} \)) x 100 = \( \frac{3 \times 100}{150} \) = \( \frac{300}{150} \) = 2% chance to win the raffle.

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

\( \frac{45\sqrt{24}}{9\sqrt{6}} \)
\( \frac{45}{9} \) \( \sqrt{\frac{24}{6}} \)
5 \( \sqrt{4} \)

The factors of 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48] and the factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]. They share 5 factors [1, 2, 4, 8, 16] making 16 the greatest factor 48 and 80 have in common.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{5!}{2!} \)
\( \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{5 \times 4 \times 3}{1} \)
\( 5 \times 4 \times 3 \)
60