The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 75% the radius (and, consequently, the total area) increases by \( \frac{75\text{%}}{2} \) = 37\(\frac{1}{2}\)%

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

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

So, in an average hour, the machine will produce 6 - 0.54 = 5.46 error free parts.

The machine ran for 24 - 8 = 16 hours yesterday so you would expect that 16 x 5.46 = 87.4 error free parts were produced yesterday.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50] and the first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 5 and 9 share.

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

\( \frac{9 x 9}{5 x 9} \) - \( \frac{8 x 5}{9 x 5} \)

\( \frac{81}{45} \) - \( \frac{40}{45} \)

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

\( \frac{81 - 40}{45} \) = \( \frac{41}{45} \) = \(\frac{41}{45}\)

To divide terms with exponents, the base of both exponents must be the same. In this case they are so divide the coefficients and subtract the exponents:

\( \frac{-6z^6}{8z^2} \)
\( \frac{-6}{8} \) z^{(6 - 2)}
-\(\frac{3}{4}\)z⁴

To take the square root of a fraction, break the fraction into two separate roots then calculate the square root of the numerator and denominator separately:

\( \sqrt{\frac{49}{25}} \)
\( \frac{\sqrt{49}}{\sqrt{25}} \)
\( \frac{\sqrt{7^2}}{\sqrt{5^2}} \)
\( \frac{7}{5} \)
1\(\frac{2}{5}\)