The factors of a number are all positive integers that divide evenly into the number. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The amount of flour you need is (2\(\frac{3}{8}\) - \(\frac{5}{8}\)) cups. Rewrite the quantities so they share a common denominator and subtract:

(\( \frac{19}{8} \) - \( \frac{5}{8} \)) cups
\( \frac{14}{8} \) cups
1\(\frac{3}{4}\) cups

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

\( \frac{4}{100} \) x 10 = \( \frac{4 \times 10}{100} \) = \( \frac{40}{100} \) = 0.4 errors per hour

So, in an average hour, the machine will produce 10 - 0.4 = 9.6 error free parts.

The machine ran for 24 - 4 = 20 hours yesterday so you would expect that 20 x 9.6 = 192 error free parts were produced yesterday.

To add 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 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 share.

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

\( \frac{2 x 2}{6 x 2} \) + \( \frac{3 x 1}{12 x 1} \)

\( \frac{4}{12} \) + \( \frac{3}{12} \)

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

\( \frac{4 + 3}{12} \) = \( \frac{7}{12} \) = \(\frac{7}{12}\)

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{81}{49}} \)
\( \frac{\sqrt{81}}{\sqrt{49}} \)
\( \frac{\sqrt{9^2}}{\sqrt{7^2}} \)
\( \frac{9}{7} \)
1\(\frac{2}{7}\)