If a single cook can bake 3 large cakes per hour and the kitchen is available for 4 hours, a single cook can bake 3 x 4 = 12 large cakes during that time. 34 large cakes are needed for the party so \( \frac{34}{12} \) = 2\(\frac{5}{6}\) cooks are needed to bake the required number of large cakes.

If a single cook can bake 20 small cakes per hour and the kitchen is available for 4 hours, a single cook can bake 20 x 4 = 80 small cakes during that time. 120 small cakes are needed for the party so \( \frac{120}{80} \) = 1\(\frac{1}{2}\) cooks are needed to bake the required number of small cakes.

Because you can't employ a fractional cook, round the number of cooks needed for each type of cake up to the next whole number resulting in 3 + 2 = 5 cooks.

To find the average of these 4 numbers add them together then divide by 4:

\( \frac{13 + 11 + 13 + 11}{4} \) = \( \frac{48}{4} \) = 12

The number of students taking German or Spanish is 10 + 15 = 25. Of that group of 25, 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 25 - 2 = 23 who are taking at least one language. 31 - 23 = 8 students who are not taking either language.

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{36}{81}} \)
\( \frac{\sqrt{36}}{\sqrt{81}} \)
\( \frac{\sqrt{6^2}}{\sqrt{9^2}} \)
\(\frac{2}{3}\)

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 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]. They share 8 factors [1, 2, 3, 4, 6, 8, 12, 24] making 24 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{48} \) = \( \frac{\frac{24}{24}}{\frac{48}{24}} \) = \( \frac{1}{2} \)