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

3\( \sqrt{6} \) x 2\( \sqrt{3} \)
(3 x 2)\( \sqrt{6 \times 3} \)
6\( \sqrt{18} \)

Now we need to simplify the radical:

6\( \sqrt{18} \)
6\( \sqrt{2 \times 9} \)
6\( \sqrt{2 \times 3^2} \)
(6)(3)\( \sqrt{2} \)
18\( \sqrt{2} \)

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

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:

\( \frac{-4y^9}{5y^3} \)
\( \frac{-4}{5} \) y^{(9 - 3)}
-\(\frac{4}{5}\)y⁶

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 65% the radius (and, consequently, the total area) increases by \( \frac{65\text{%}}{2} \) = 32\(\frac{1}{2}\)%