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

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{4}{6} \) x \( \frac{1}{5} \) = \( \frac{4 x 1}{6 x 5} \) = \( \frac{4}{30} \) = \(\frac{2}{15}\)

The number of students taking German or Spanish is 11 + 12 = 23. Of that group of 23, 7 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 - 7 = 16 who are taking at least one language. 26 - 16 = 10 students who are not taking either language.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

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

\( \frac{7 x 2}{4 x 2} \) - \( \frac{4 x 1}{8 x 1} \)

\( \frac{14}{8} \) - \( \frac{4}{8} \)

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

\( \frac{14 - 4}{8} \) = \( \frac{10}{8} \) = 1\(\frac{1}{4}\)

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:

-8a⁷ - 6a⁷
(-8 - 6)a⁷
-14a⁷