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

5\( \sqrt{9} \) x 3\( \sqrt{7} \)
(5 x 3)\( \sqrt{9 \times 7} \)
15\( \sqrt{63} \)

Now we need to simplify the radical:

15\( \sqrt{63} \)
15\( \sqrt{7 \times 9} \)
15\( \sqrt{7 \times 3^2} \)
(15)(3)\( \sqrt{7} \)
45\( \sqrt{7} \)

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

39,000 fans x \( \frac{4}{5} \) = \( \frac{156000}{5} \) = 31,200 fans.

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

4\( \sqrt{80} \) - 2\( \sqrt{5} \)
4\( \sqrt{16 \times 5} \) - 2\( \sqrt{5} \)
4\( \sqrt{4^2 \times 5} \) - 2\( \sqrt{5} \)
(4)(4)\( \sqrt{5} \) - 2\( \sqrt{5} \)
16\( \sqrt{5} \) - 2\( \sqrt{5} \)

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

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 75% the radius (and, consequently, the total area) increases by \( \frac{75\text{%}}{2} \) = 37\(\frac{1}{2}\)%

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.