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.

The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [21, 42, 63, 84] making 21 the smallest multiple 3 and 7 have in common.

You have 6 out of the total of 300 raffle tickets sold so you have a (\( \frac{6}{300} \)) x 100 = \( \frac{6 \times 100}{300} \) = \( \frac{600}{300} \) = 2% chance to win the raffle.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 70mph \times 1h \)
70 miles

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

4\( \sqrt{3} \) x 5\( \sqrt{9} \)
(4 x 5)\( \sqrt{3 \times 9} \)
20\( \sqrt{27} \)

Now we need to simplify the radical:

20\( \sqrt{27} \)
20\( \sqrt{3 \times 9} \)
20\( \sqrt{3 \times 3^2} \)
(20)(3)\( \sqrt{3} \)
60\( \sqrt{3} \)