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 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 35% the radius (and, consequently, the total area) increases by \( \frac{35\text{%}}{2} \) = 17\(\frac{1}{2}\)%

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

4\( \sqrt{9} \) x 6\( \sqrt{8} \)
(4 x 6)\( \sqrt{9 \times 8} \)
24\( \sqrt{72} \)

Now we need to simplify the radical:

24\( \sqrt{72} \)
24\( \sqrt{2 \times 36} \)
24\( \sqrt{2 \times 6^2} \)
(24)(6)\( \sqrt{2} \)
144\( \sqrt{2} \)

A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

35,000 fans x \( \frac{3}{4} \) = \( \frac{105000}{4} \) = 26,250 fans.

The greatest common factor (GCF) is the greatest factor that divides two integers.