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

8\( \sqrt{3} \) x 8\( \sqrt{4} \)
(8 x 8)\( \sqrt{3 \times 4} \)
64\( \sqrt{12} \)

Now we need to simplify the radical:

64\( \sqrt{12} \)
64\( \sqrt{3 \times 4} \)
64\( \sqrt{3 \times 2^2} \)
(64)(2)\( \sqrt{3} \)
128\( \sqrt{3} \)

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 10 = \( \frac{5 \times 10}{100} \) = \( \frac{50}{100} \) = 0.5 errors per hour

So, in an average hour, the machine will produce 10 - 0.5 = 9.5 error free parts.

The machine ran for 24 - 8 = 16 hours yesterday so you would expect that 16 x 9.5 = 152 error free parts were produced yesterday.

The factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80] and the factors of 28 are [1, 2, 4, 7, 14, 28]. They share 3 factors [1, 2, 4] making 4 the greatest factor 80 and 28 have in common.

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:

\( \frac{4z^6}{2z^2} \)
\( \frac{4}{2} \) z^{(6 - 2)}
2z⁴

If 55% of the town's 40,000 voters cast ballots the number of votes cast is:

(\( \frac{55}{100} \)) x 40,000 = \( \frac{2,200,000}{100} \) = 22,000

The mayor got 56% of the votes cast which is:

(\( \frac{56}{100} \)) x 22,000 = \( \frac{1,232,000}{100} \) = 12,320 votes.