The number of students taking German or Spanish is 12 + 11 = 23. Of that group of 23, 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 23 - 2 = 21 who are taking at least one language. 27 - 21 = 6 students who are not taking either language.

The yearly interest charged on this loan is the annual interest rate multiplied by the amount borrowed:

interest = annual interest rate x loan amount

i = (\( \frac{6}{100} \)) x $600
i = 0.08 x $600
i = $48

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

The equation for this sequence is:

a_n = a_n-1 + 4(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 4(6 - 1)
a₆ = 41 + 4(5)
a₆ = 61

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] 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 [35, 70] making 35 the smallest multiple 5 and 7 share.

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

\( \frac{5 x 7}{5 x 7} \) + \( \frac{6 x 5}{7 x 5} \)

\( \frac{35}{35} \) + \( \frac{30}{35} \)

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

\( \frac{35 + 30}{35} \) = \( \frac{65}{35} \) = 1\(\frac{6}{7}\)