To simplify a radical, factor out the perfect squares:

\( \sqrt{75} \)
\( \sqrt{25 \times 3} \)
\( \sqrt{5^2 \times 3} \)
5\( \sqrt{3} \)

You have 9 out of the total of 100 raffle tickets sold so you have a (\( \frac{9}{100} \)) x 100 = \( \frac{9 \times 100}{100} \) = \( \frac{900}{100} \) = 9% chance to win the raffle.

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{2}{7} \) x \( \frac{4}{9} \) = \( \frac{2 x 4}{7 x 9} \) = \( \frac{8}{63} \) = \(\frac{8}{63}\)

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40]. The first few multiples they share are [4, 8, 12, 16, 20] making 4 the smallest multiple 2 and 4 share.

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

\( \frac{7 x 2}{2 x 2} \) - \( \frac{4 x 1}{4 x 1} \)

\( \frac{14}{4} \) - \( \frac{4}{4} \)

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

\( \frac{14 - 4}{4} \) = \( \frac{10}{4} \) = 2\(\frac{1}{2}\)

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

\( \frac{7}{100} \) x 8 = \( \frac{7 \times 8}{100} \) = \( \frac{56}{100} \) = 0.56 errors per hour

So, in an average hour, the machine will produce 8 - 0.56 = 7.4399999999999995 error free parts.

The machine ran for 24 - 7 = 17 hours yesterday so you would expect that 17 x 7.4399999999999995 = 126.5 error free parts were produced yesterday.