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

\( \frac{1}{9} \) x \( \frac{1}{9} \) = \( \frac{1 x 1}{9 x 9} \) = \( \frac{1}{81} \) = \(\frac{1}{81}\)

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

4y³ x 2y⁷
(4 x 2)y^{(3 + 7)}
8y¹⁰

The number of students taking German or Spanish is 5 + 10 = 15. Of that group of 15, 3 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 15 - 3 = 12 who are taking at least one language. 27 - 12 = 15 students who are not taking either language.

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:

7y⁷ + 7y⁷
(7 + 7)y⁷
14y⁷

To simplify this fraction, first find the greatest common factor between them. The factors of 24 are [1, 2, 3, 4, 6, 8, 12, 24] and the factors of 56 are [1, 2, 4, 7, 8, 14, 28, 56]. They share 4 factors [1, 2, 4, 8] making 8 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{56} \) = \( \frac{\frac{24}{8}}{\frac{56}{8}} \) = \( \frac{3}{7} \)