The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

You have 13 out of the total of 150 raffle tickets sold so you have a (\( \frac{13}{150} \)) x 100 = \( \frac{13 \times 100}{150} \) = \( \frac{1300}{150} \) = 9% chance to win the raffle.

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

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

2\( \sqrt{63} \) - 2\( \sqrt{7} \)
2\( \sqrt{9 \times 7} \) - 2\( \sqrt{7} \)
2\( \sqrt{3^2 \times 7} \) - 2\( \sqrt{7} \)
(2)(3)\( \sqrt{7} \) - 2\( \sqrt{7} \)
6\( \sqrt{7} \) - 2\( \sqrt{7} \)

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

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 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 2 and 8 share.

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

\( \frac{9 x 4}{2 x 4} \) + \( \frac{5 x 1}{8 x 1} \)

\( \frac{36}{8} \) + \( \frac{5}{8} \)

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

\( \frac{36 + 5}{8} \) = \( \frac{41}{8} \) = 5\(\frac{1}{8}\)