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

8\( \sqrt{2} \) x 4\( \sqrt{9} \)
(8 x 4)\( \sqrt{2 \times 9} \)
32\( \sqrt{18} \)

Now we need to simplify the radical:

32\( \sqrt{18} \)
32\( \sqrt{2 \times 9} \)
32\( \sqrt{2 \times 3^2} \)
(32)(3)\( \sqrt{2} \)
96\( \sqrt{2} \)

To simplify a radical, factor out the perfect squares:

\( \sqrt{75} \)
\( \sqrt{25 \times 3} \)
\( \sqrt{5^2 \times 3} \)
5\( \sqrt{3} \)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{8 x 2}{2 x 2} \) + \( \frac{7 x 1}{4 x 1} \)

\( \frac{16}{4} \) + \( \frac{7}{4} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{16 + 7}{4} \) = \( \frac{23}{4} \) = 5\(\frac{3}{4}\)

You have 10 out of the total of 350 raffle tickets sold so you have a (\( \frac{10}{350} \)) x 100 = \( \frac{10 \times 100}{350} \) = \( \frac{1000}{350} \) = 3% chance to win the raffle.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.