To simplify this fraction, first find the greatest common factor between them. The factors of 28 are [1, 2, 4, 7, 14, 28] and the factors of 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{28}{48} \) = \( \frac{\frac{28}{4}}{\frac{48}{4}} \) = \( \frac{7}{12} \)

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

\( \frac{6}{100} \) x 7 = \( \frac{6 \times 7}{100} \) = \( \frac{42}{100} \) = 0.42 errors per hour

So, in an average hour, the machine will produce 7 - 0.42 = 6.58 error free parts.

The machine ran for 24 - 3 = 21 hours yesterday so you would expect that 21 x 6.58 = 138.2 error free parts were produced yesterday.

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 radicals, multiply the coefficients and radicands separately:

4\( \sqrt{2} \) x 6\( \sqrt{9} \)
(4 x 6)\( \sqrt{2 \times 9} \)
24\( \sqrt{18} \)

Now we need to simplify the radical:

24\( \sqrt{18} \)
24\( \sqrt{2 \times 9} \)
24\( \sqrt{2 \times 3^2} \)
(24)(3)\( \sqrt{2} \)
72\( \sqrt{2} \)

The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [40, 80] making 40 the smallest multiple 8 and 10 have in common.