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:

\( \frac{-6a^6}{9a^3} \)
\( \frac{-6}{9} \) a^{(6 - 3)}
-\(\frac{2}{3}\)a³

The equation for this sequence is:

a_n = a_n-1 + 6

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 6
a₆ = 25 + 6
a₆ = 31

The factors of 44 are [1, 2, 4, 11, 22, 44] and the factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40]. They share 3 factors [1, 2, 4] making 4 the greatest factor 44 and 40 have in common.

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

If a single cook can bake 18 small cakes per hour and the kitchen is available for 3 hours, a single cook can bake 18 x 3 = 54 small cakes during that time. 500 small cakes are needed for the party so \( \frac{500}{54} \) = 9\(\frac{7}{27}\) 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 6 + 10 = 16 cooks.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{4!}{6!} \)
\( \frac{4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5} \)
\( \frac{1}{30} \)