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

-1z⁵ + 2z⁵
(-1 + 2)z⁵
z⁵

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{7z^6}{9z^2} \)
\( \frac{7}{9} \) z^{(6 - 2)}
\(\frac{7}{9}\)z⁴

The number of students taking German or Spanish is 5 + 12 = 17. Of that group of 17, 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 17 - 4 = 13 who are taking at least one language. 24 - 13 = 11 students who are not taking either language.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 55mph \times 9h \)
495 miles

To subtract 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 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 share.

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

\( \frac{3 x 9}{5 x 9} \) - \( \frac{5 x 5}{9 x 5} \)

\( \frac{27}{45} \) - \( \frac{25}{45} \)

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

\( \frac{27 - 25}{45} \) = \( \frac{2}{45} \) = \(\frac{2}{45}\)