The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 11 are [11, 22, 33, 44, 55, 66, 77, 88, 99]. The first few multiples they share are [33, 66, 99] making 33 the smallest multiple 3 and 11 share.

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

\( \frac{2 x 11}{3 x 11} \) + \( \frac{4 x 3}{11 x 3} \)

\( \frac{22}{33} \) + \( \frac{12}{33} \)

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

\( \frac{22 + 12}{33} \) = \( \frac{34}{33} \) = 1\(\frac{1}{33}\)

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

4\( \sqrt{18} \) - 8\( \sqrt{2} \)
4\( \sqrt{9 \times 2} \) - 8\( \sqrt{2} \)
4\( \sqrt{3^2 \times 2} \) - 8\( \sqrt{2} \)
(4)(3)\( \sqrt{2} \) - 8\( \sqrt{2} \)
12\( \sqrt{2} \) - 8\( \sqrt{2} \)

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

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.

If 60% of the town's 39,000 voters cast ballots the number of votes cast is:

(\( \frac{60}{100} \)) x 39,000 = \( \frac{2,340,000}{100} \) = 23,400

The mayor got 80% of the votes cast which is:

(\( \frac{80}{100} \)) x 23,400 = \( \frac{1,872,000}{100} \) = 18,720 votes.