The number of students taking German or Spanish is 13 + 14 = 27. Of that group of 27, 5 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 27 - 5 = 22 who are taking at least one language. 32 - 22 = 10 students who are not taking either language.

To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

-3y⁴ x 4y⁵
(-3 x 4)y^{(4 + 5)}
-12y⁹

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

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

Now we need to simplify the radical:

54\( \sqrt{72} \)
54\( \sqrt{2 \times 36} \)
54\( \sqrt{2 \times 6^2} \)
(54)(6)\( \sqrt{2} \)
324\( \sqrt{2} \)

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 76 are [1, 2, 4, 19, 38, 76]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{76} \) = \( \frac{\frac{40}{4}}{\frac{76}{4}} \) = \( \frac{10}{19} \)

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

-2z⁴ + 8z⁴
(-2 + 8)z⁴
6z⁴