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 6 = \( \frac{7 \times 6}{100} \) = \( \frac{42}{100} \) = 0.42 errors per hour

So, in an average hour, the machine will produce 6 - 0.42 = 5.58 error free parts.

The machine ran for 24 - 7 = 17 hours yesterday so you would expect that 17 x 5.58 = 94.9 error free parts were produced yesterday.

The number of students taking German or Spanish is 8 + 7 = 15. Of that group of 15, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 15 - 5 = 10 who are taking at least one language. 22 - 10 = 12 students who are not taking either language.

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

7c⁷ x 4c⁴
(7 x 4)c^{(7 + 4)}
28c¹¹

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

To simplify a radical, factor out the perfect squares:

\( \sqrt{32} \)
\( \sqrt{16 \times 2} \)
\( \sqrt{4^2 \times 2} \)
4\( \sqrt{2} \)