To add these radicals together their radicands must be the same:

8\( \sqrt{20} \) + 6\( \sqrt{5} \)
8\( \sqrt{4 \times 5} \) + 6\( \sqrt{5} \)
8\( \sqrt{2^2 \times 5} \) + 6\( \sqrt{5} \)
(8)(2)\( \sqrt{5} \) + 6\( \sqrt{5} \)
16\( \sqrt{5} \) + 6\( \sqrt{5} \)

Now that the radicands are identical, you can add them together:

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

9\( \sqrt{9} \) x 8\( \sqrt{3} \)
(9 x 8)\( \sqrt{9 \times 3} \)
72\( \sqrt{27} \)

Now we need to simplify the radical:

72\( \sqrt{27} \)
72\( \sqrt{3 \times 9} \)
72\( \sqrt{3 \times 3^2} \)
(72)(3)\( \sqrt{3} \)
216\( \sqrt{3} \)

To add 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 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{2 x 5}{2 x 5} \) + \( \frac{6 x 1}{10 x 1} \)

\( \frac{10}{10} \) + \( \frac{6}{10} \)

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

\( \frac{10 + 6}{10} \) = \( \frac{16}{10} \) = 1\(\frac{3}{5}\)

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

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.