The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.

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

\( \frac{3}{100} \) x 7 = \( \frac{3 \times 7}{100} \) = \( \frac{21}{100} \) = 0.21 errors per hour

So, in an average hour, the machine will produce 7 - 0.21 = 6.79 error free parts.

The machine ran for 24 - 4 = 20 hours yesterday so you would expect that 20 x 6.79 = 135.8 error free parts were produced yesterday.

The number of students taking German or Spanish is 14 + 14 = 28. Of that group of 28, 2 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 28 - 2 = 26 who are taking at least one language. 33 - 26 = 7 students who are not taking either language.

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

2\( \sqrt{63} \) + 6\( \sqrt{7} \)
2\( \sqrt{9 \times 7} \) + 6\( \sqrt{7} \)
2\( \sqrt{3^2 \times 7} \) + 6\( \sqrt{7} \)
(2)(3)\( \sqrt{7} \) + 6\( \sqrt{7} \)
6\( \sqrt{7} \) + 6\( \sqrt{7} \)

Now that the radicands are identical, you can add them together:

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for time:

time = \( \frac{\text{distance}}{\text{speed}} \)
time = \( \frac{90mi}{45mph} \)
2 hours