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.

A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-y^5}{8y^3} \)
\( \frac{-1}{8} \) y^{(5 - 3)}
-\(\frac{1}{8}\)y²

If 71% of the town's 30,000 voters cast ballots the number of votes cast is:

(\( \frac{71}{100} \)) x 30,000 = \( \frac{2,130,000}{100} \) = 21,300

The mayor got 78% of the votes cast which is:

(\( \frac{78}{100} \)) x 21,300 = \( \frac{1,661,400}{100} \) = 16,614 votes.

The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 70% the radius (and, consequently, the total area) increases by \( \frac{70\text{%}}{2} \) = 35%