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:

-9y⁷ + 6y⁷
(-9 + 6)y⁷
-3y⁷

To subtract 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 16 are [16, 32, 48, 64, 80, 96]. The first few multiples they share are [16, 32, 48, 64, 80] making 16 the smallest multiple 8 and 16 share.

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

\( \frac{8 x 2}{8 x 2} \) - \( \frac{3 x 1}{16 x 1} \)

\( \frac{16}{16} \) - \( \frac{3}{16} \)

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

\( \frac{16 - 3}{16} \) = \( \frac{13}{16} \) = \(\frac{13}{16}\)

The ratio of football cards to baseball cards is 9:2 and the ratio of baseball cards to basketball cards is 9:1. To solve this problem, we need the baseball card side of each ratio to be equal so we need to rewrite the ratios in terms of a common number of baseball cards. (Think of this like finding the common denominator when adding fractions.) The ratio of football to baseball cards can also be written as 81:18 and the ratio of baseball cards to basketball cards as 18:2. So, the ratio of football cards to basketball cards is football:baseball, baseball:basketball or 81:18, 18:2 which reduces to 81:2.

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 15 workers at the company now and that's enough to staff 5 crews so there are \( \frac{15}{5} \) = 3 workers on a crew. 7 crews are needed for the busy season which, at 3 workers per crew, means that the roofing company will need 7 x 3 = 21 total workers to staff the crews during the busy season. The company already employs 15 workers so they need to add 21 - 15 = 6 new staff for the busy season.

A factor is a positive integer that divides evenly into a given number. For example, the factors of 8 are 1, 2, 4, and 8.