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

4\( \sqrt{8} \) x 4\( \sqrt{5} \)
(4 x 4)\( \sqrt{8 \times 5} \)
16\( \sqrt{40} \)

Now we need to simplify the radical:

16\( \sqrt{40} \)
16\( \sqrt{10 \times 4} \)
16\( \sqrt{10 \times 2^2} \)
(16)(2)\( \sqrt{10} \)
32\( \sqrt{10} \)

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.

You have 12 out of the total of 200 raffle tickets sold so you have a (\( \frac{12}{200} \)) x 100 = \( \frac{12 \times 100}{200} \) = \( \frac{1200}{200} \) = 6% chance to win the raffle.

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.