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, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

A prime number is an integer greater than 1 that has no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

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

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

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

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 18 meters so the equation becomes: 2w + 2h = 18.

Putting these two equations together and solving for width (w):

2w + 2h = 18
w + h = \( \frac{18}{2} \)
w + h = 9
w = 9 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 9 - 2w
3w = 9
w = \( \frac{9}{3} \)
w = 3

Since h = 2w that makes h = (2 x 3) = 6 and the area = h x w = 3 x 6 = 18 m²

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 5 = \( \frac{7 \times 5}{100} \) = \( \frac{35}{100} \) = 0.35 errors per hour

So, in an average hour, the machine will produce 5 - 0.35 = 4.65 error free parts.

The machine ran for 24 - 9 = 15 hours yesterday so you would expect that 15 x 4.65 = 69.8 error free parts were produced yesterday.