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:

\( \frac{-7z^5}{5z^2} \)
\( \frac{-7}{5} \) z^{(5 - 2)}
-1\(\frac{2}{5}\)z³

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{2}{100} \) x 10 = \( \frac{2 \times 10}{100} \) = \( \frac{20}{100} \) = 0.2 errors per hour

So, in an average hour, the machine will produce 10 - 0.2 = 9.8 error free parts.

The machine ran for 24 - 7 = 17 hours yesterday so you would expect that 17 x 9.8 = 166.6 error free parts were produced yesterday.

The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{6 x 5}{9 x 5} \) + \( \frac{9 x 3}{15 x 3} \)

\( \frac{30}{45} \) + \( \frac{27}{45} \)

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

\( \frac{30 + 27}{45} \) = \( \frac{57}{45} \) = 1\(\frac{4}{15}\)

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.