To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] 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:

\( \frac{40}{56} \) = \( \frac{\frac{40}{8}}{\frac{56}{8}} \) = \( \frac{5}{7} \)

To divide terms with radicals, divide the coefficients and radicands separately:

\( \frac{42\sqrt{16}}{6\sqrt{4}} \)
\( \frac{42}{6} \) \( \sqrt{\frac{16}{4}} \)
7 \( \sqrt{4} \)

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

\( \frac{9}{100} \) x 6 = \( \frac{9 \times 6}{100} \) = \( \frac{54}{100} \) = 0.54 errors per hour

So, in an average hour, the machine will produce 6 - 0.54 = 5.46 error free parts.

The machine ran for 24 - 3 = 21 hours yesterday so you would expect that 21 x 5.46 = 114.7 error free parts were produced yesterday.

If 58% of the town's 33,000 voters cast ballots the number of votes cast is:

(\( \frac{58}{100} \)) x 33,000 = \( \frac{1,914,000}{100} \) = 19,140

The mayor got 89% of the votes cast which is:

(\( \frac{89}{100} \)) x 19,140 = \( \frac{1,703,460}{100} \) = 17,035 votes.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [30, 60, 90] making 30 the smallest multiple 6 and 10 share.

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

\( \frac{4 x 5}{6 x 5} \) - \( \frac{6 x 3}{10 x 3} \)

\( \frac{20}{30} \) - \( \frac{18}{30} \)

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

\( \frac{20 - 18}{30} \) = \( \frac{2}{30} \) = \(\frac{1}{15}\)