To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so subtract the coefficients and retain the base and exponent:

-4x⁶ - 8x⁶
(-4 - 8)x⁶
-12x⁶

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 2 and 8 share.

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

\( \frac{2 x 4}{2 x 4} \) - \( \frac{4 x 1}{8 x 1} \)

\( \frac{8}{8} \) - \( \frac{4}{8} \)

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

\( \frac{8 - 4}{8} \) = \( \frac{4}{8} \) = \(\frac{1}{2}\)

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

\( \frac{6}{100} \) x 9 = \( \frac{6 \times 9}{100} \) = \( \frac{54}{100} \) = 0.54 errors per hour

So, in an average hour, the machine will produce 9 - 0.54 = 8.46 error free parts.

The machine ran for 24 - 2 = 22 hours yesterday so you would expect that 22 x 8.46 = 186.1 error free parts were produced yesterday.

The equation for this sequence is:

a_n = a_n-1 + 8

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 8
a₆ = 33 + 8
a₆ = 41

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.