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:

9z⁷ x z³
(9 x 1)z^{(7 + 3)}
9z¹⁰

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

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

\( \frac{2 x 3}{8 x 3} \) + \( \frac{7 x 2}{12 x 2} \)

\( \frac{6}{24} \) + \( \frac{14}{24} \)

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

\( \frac{6 + 14}{24} \) = \( \frac{20}{24} \) = \(\frac{5}{6}\)

1

In order to find how many additional workers are needed to staff the extra crews you first need to calculate how many workers are on a crew. There are 6 workers at the company now and that's enough to staff 2 crews so there are \( \frac{6}{2} \) = 3 workers on a crew. 5 crews are needed for the busy season which, at 3 workers per crew, means that the roofing company will need 5 x 3 = 15 total workers to staff the crews during the busy season. The company already employs 6 workers so they need to add 15 - 6 = 9 new staff for the busy season.

To fill a 12 gallon tank exactly halfway you'll need 6 gallons of fuel. Each fuel can holds 2 gallons so:

cans = \( \frac{6 \text{ gallons}}{2 \text{ gallons}} \) = 3