To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

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

\( \frac{6 x 2}{2 x 2} \) + \( \frac{4 x 1}{4 x 1} \)

\( \frac{12}{4} \) + \( \frac{4}{4} \)

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

\( \frac{12 + 4}{4} \) = \( \frac{16}{4} \) = 4

To raise a term with an exponent to another exponent, retain the base and multiply the exponents:

The number of students taking German or Spanish is 10 + 9 = 19. Of that group of 19, 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 19 - 2 = 17 who are taking at least one language. 28 - 17 = 11 students who are not taking either language.

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 first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [21, 42, 63, 84] making 21 the smallest multiple 3 and 7 have in common.