To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 7 are [7, 14, 21, 28, 35, 42, 49, 56, 63, 70]. The first few multiples they share are [35, 70] making 35 the smallest multiple 5 and 7 share.

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

\( \frac{6 x 7}{5 x 7} \) + \( \frac{2 x 5}{7 x 5} \)

\( \frac{42}{35} \) + \( \frac{10}{35} \)

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

\( \frac{42 + 10}{35} \) = \( \frac{52}{35} \) = 1\(\frac{17}{35}\)

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

\( \frac{7}{100} \) x 10 = \( \frac{7 \times 10}{100} \) = \( \frac{70}{100} \) = 0.7 errors per hour

So, in an average hour, the machine will produce 10 - 0.7 = 9.3 error free parts.

The machine ran for 24 - 9 = 15 hours yesterday so you would expect that 15 x 9.3 = 139.5 error free parts were produced yesterday.

If 82% of the town's 33,000 voters cast ballots the number of votes cast is:

(\( \frac{82}{100} \)) x 33,000 = \( \frac{2,706,000}{100} \) = 27,060

The mayor got 52% of the votes cast which is:

(\( \frac{52}{100} \)) x 27,060 = \( \frac{1,407,120}{100} \) = 14,071 votes.

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

2\( \sqrt{8} \) - 5\( \sqrt{2} \)
2\( \sqrt{4 \times 2} \) - 5\( \sqrt{2} \)
2\( \sqrt{2^2 \times 2} \) - 5\( \sqrt{2} \)
(2)(2)\( \sqrt{2} \) - 5\( \sqrt{2} \)
4\( \sqrt{2} \) - 5\( \sqrt{2} \)

Now that the radicands are identical, you can subtract them:

To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

7z⁴ + 5z⁴
(7 + 5)z⁴
12z⁴