The number of students taking German or Spanish is 15 + 10 = 25. Of that group of 25, 2 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 25 - 2 = 23 who are taking at least one language. 31 - 23 = 8 students who are not taking either language.

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%

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

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

Now that the radicands are identical, you can add them together:

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{25}{25}} \)
\( \frac{\sqrt{25}}{\sqrt{25}} \)
\( \frac{\sqrt{5^2}}{\sqrt{5^2}} \)
1

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.