Diane scored 82% on the test meaning she earned 82% of the possible points on the test. There were 150 possible points on the test so she earned 150 x 0.82 = 123 points. Each question is worth 3 points so she got \( \frac{123}{3} \) = 41 questions right.

You have 8 out of the total of 200 raffle tickets sold so you have a (\( \frac{8}{200} \)) x 100 = \( \frac{8 \times 100}{200} \) = \( \frac{800}{200} \) = 4% chance to win the raffle.

To simplify this fraction, first find the greatest common factor between them. The factors of 36 are [1, 2, 3, 4, 6, 9, 12, 18, 36] and the factors of 52 are [1, 2, 4, 13, 26, 52]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{36}{52} \) = \( \frac{\frac{36}{4}}{\frac{52}{4}} \) = \( \frac{9}{13} \)

To simplify a radical, factor out the perfect squares:

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

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

\( \frac{40\sqrt{14}}{8\sqrt{2}} \)
\( \frac{40}{8} \) \( \sqrt{\frac{14}{2}} \)
5 \( \sqrt{7} \)