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:

\( \sqrt{\frac{36}{16}} \)
\( \frac{\sqrt{36}}{\sqrt{16}} \)
\( \frac{\sqrt{6^2}}{\sqrt{4^2}} \)
\( \frac{6}{4} \)
1\(\frac{1}{2}\)

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

\( \frac{5}{100} \) x 9 = \( \frac{5 \times 9}{100} \) = \( \frac{45}{100} \) = 0.45 errors per hour

So, in an average hour, the machine will produce 9 - 0.45 = 8.55 error free parts.

The machine ran for 24 - 3 = 21 hours yesterday so you would expect that 21 x 8.55 = 179.6 error free parts were produced yesterday.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for time:

time = \( \frac{\text{distance}}{\text{speed}} \)
time = \( \frac{520mi}{65mph} \)
8 hours

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:

-4c⁴ - 5c⁴
(-4 - 5)c⁴
-9c⁴

By buying two shirts, Roger will save $31 x \( \frac{25}{100} \) = \( \frac{$31 x 25}{100} \) = \( \frac{$775}{100} \) = $7.75 on the second shirt.