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:

-3x⁷ - 1x⁷
(-3 - 1)x⁷
-4x⁷

The number of students taking German or Spanish is 10 + 10 = 20. Of that group of 20, 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 20 - 2 = 18 who are taking at least one language. 33 - 18 = 15 students who are not taking either language.

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 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{9 x 5}{2 x 5} \) + \( \frac{6 x 1}{10 x 1} \)

\( \frac{45}{10} \) + \( \frac{6}{10} \)

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

\( \frac{45 + 6}{10} \) = \( \frac{51}{10} \) = 5\(\frac{1}{10}\)

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

3 + (5 + 3) ÷ 5 x 4 - 2²
P: 3 + (8) ÷ 5 x 4 - 2²
E: 3 + 8 ÷ 5 x 4 - 4
MD: 3 + \( \frac{8}{5} \) x 4 - 4
MD: 3 + \( \frac{32}{5} \) - 4
AS: \( \frac{15}{5} \) + \( \frac{32}{5} \) - 4
AS: \( \frac{47}{5} \) - 4
AS: \( \frac{47 - 20}{5} \)
\( \frac{27}{5} \)
5\(\frac{2}{5}\)