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.

The number of students taking German or Spanish is 14 + 5 = 19. Of that group of 19, 4 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 19 - 4 = 15 who are taking at least one language. 22 - 15 = 7 students who are not taking either language.

To simplify a radical, factor out the perfect squares:

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

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{5 x 3}{8 x 3} \) + \( \frac{6 x 2}{12 x 2} \)

\( \frac{15}{24} \) + \( \frac{12}{24} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{15 + 12}{24} \) = \( \frac{27}{24} \) = 1\(\frac{1}{8}\)

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:

-6a⁷ x 7a³
(-6 x 7)a^{(7 + 3)}
-42a¹⁰