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 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:

\( \frac{8 x 3}{8 x 3} \) - \( \frac{6 x 2}{12 x 2} \)

\( \frac{24}{24} \) - \( \frac{12}{24} \)

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

\( \frac{24 - 12}{24} \) = \( \frac{12}{24} \) = \(\frac{1}{2}\)

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:

-6z⁵ + 7z⁵
(-6 + 7)z⁵
z⁵

A ratio of 2:1 means that there are 2 home fans for every one visiting fan. So, of every 3 fans, 2 are home fans and \( \frac{2}{3} \) of every fan in the stadium is a home fan:

42,000 fans x \( \frac{2}{3} \) = \( \frac{84000}{3} \) = 28,000 fans.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

The amount of flour you need is (2\(\frac{3}{4}\) - \(\frac{3}{4}\)) cups. Rewrite the quantities so they share a common denominator and subtract:

(\( \frac{22}{8} \) - \( \frac{6}{8} \)) cups
\( \frac{16}{8} \) cups
2 cups