The greatest common factor (GCF) is the greatest factor that divides two integers.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{7 x 3}{8 x 3} \) + \( \frac{7 x 2}{12 x 2} \)

\( \frac{21}{24} \) + \( \frac{14}{24} \)

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

\( \frac{21 + 14}{24} \) = \( \frac{35}{24} \) = 1\(\frac{11}{24}\)

A factorial has the form n! and is the product of the integer (n) and all the positive integers below it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

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:

-5z⁶ x 4z³
(-5 x 4)z^{(6 + 3)}
-20z⁹