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

2\( \sqrt{2} \) x 6\( \sqrt{2} \)
(2 x 6)\( \sqrt{2 \times 2} \)
12\( \sqrt{4} \)

Now we need to simplify the radical:

12\( \sqrt{4} \)
12\( \sqrt{2^2} \)
(12)(2)
24

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{7x^7}{2x^2} \)
\( \frac{7}{2} \) x^{(7 - 2)}
3\(\frac{1}{2}\)x⁵

The factors of 16 are [1, 2, 4, 8, 16] and the factors of 20 are [1, 2, 4, 5, 10, 20]. They share 3 factors [1, 2, 4] making 4 the greatest factor 16 and 20 have in common.

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

\( \frac{5}{100} \) x 6 = \( \frac{5 \times 6}{100} \) = \( \frac{30}{100} \) = 0.3 errors per hour

So, in an average hour, the machine will produce 6 - 0.3 = 5.7 error free parts.

The machine ran for 24 - 7 = 17 hours yesterday so you would expect that 17 x 5.7 = 96.9 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:

\( \frac{4 x 2}{6 x 2} \) + \( \frac{8 x 1}{12 x 1} \)

\( \frac{8}{12} \) + \( \frac{8}{12} \)

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

\( \frac{8 + 8}{12} \) = \( \frac{16}{12} \) = 1\(\frac{1}{3}\)