The number of students taking German or Spanish is 8 + 8 = 16. Of that group of 16, 3 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 16 - 3 = 13 who are taking at least one language. 19 - 13 = 6 students who are not taking either language.

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.

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:

\( \sqrt{\frac{16}{64}} \)
\( \frac{\sqrt{16}}{\sqrt{64}} \)
\( \frac{\sqrt{4^2}}{\sqrt{8^2}} \)
\(\frac{1}{2}\)

You have 18 out of the total of 200 raffle tickets sold so you have a (\( \frac{18}{200} \)) x 100 = \( \frac{18 \times 100}{200} \) = \( \frac{1800}{200} \) = 9% chance to win the raffle.

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

-5z⁶ - 9z⁶
(-5 - 9)z⁶
-14z⁶