To simplify a radical, factor out the perfect squares:

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

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

\( \frac{8}{100} \) x 10 = \( \frac{8 \times 10}{100} \) = \( \frac{80}{100} \) = 0.8 errors per hour

So, in an average hour, the machine will produce 10 - 0.8 = 9.2 error free parts.

The machine ran for 24 - 9 = 15 hours yesterday so you would expect that 15 x 9.2 = 138 error free parts were produced yesterday.

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

\( \frac{12\sqrt{6}}{6\sqrt{3}} \)
\( \frac{12}{6} \) \( \sqrt{\frac{6}{3}} \)
2 \( \sqrt{2} \)

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

35,000 fans x \( \frac{4}{5} \) = \( \frac{140000}{5} \) = 28,000 fans.

The equation for this sequence is:

a_n = a_n-1 + 3(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 3(6 - 1)
a₆ = 31 + 3(5)
a₆ = 46