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 60% the radius (and, consequently, the total area) increases by \( \frac{60\text{%}}{2} \) = 30%

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

9\( \sqrt{32} \) - 8\( \sqrt{2} \)
9\( \sqrt{16 \times 2} \) - 8\( \sqrt{2} \)
9\( \sqrt{4^2 \times 2} \) - 8\( \sqrt{2} \)
(9)(4)\( \sqrt{2} \) - 8\( \sqrt{2} \)
36\( \sqrt{2} \) - 8\( \sqrt{2} \)

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

The yearly interest charged on this loan is the annual interest rate multiplied by the amount borrowed:

interest = annual interest rate x loan amount

i = (\( \frac{6}{100} \)) x $200
i = 0.03 x $200
i = $6

First, solve for \( \left|y - 3\right| \):

\( \left|y - 3\right| \) + 9 = 1
\( \left|y - 3\right| \) = 1 - 9
\( \left|y - 3\right| \) = -8

The value inside the absolute value brackets can be either positive or negative so (y - 3) must equal - 8 or --8 for \( \left|y - 3\right| \) to equal -8:

So, y = 11 or y = -5.