To add these radicals together their radicands must be the same:

4\( \sqrt{63} \) + 8\( \sqrt{7} \)
4\( \sqrt{9 \times 7} \) + 8\( \sqrt{7} \)
4\( \sqrt{3^2 \times 7} \) + 8\( \sqrt{7} \)
(4)(3)\( \sqrt{7} \) + 8\( \sqrt{7} \)
12\( \sqrt{7} \) + 8\( \sqrt{7} \)

Now that the radicands are identical, you can add them together:

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 60% the radius (and, consequently, the total area) increases by \( \frac{60\text{%}}{2} \) = 30%

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

8z⁵ x 4z⁶
(8 x 4)z^{(5 + 6)}
32z¹¹

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

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 6 = \( \frac{5 \times 6}{100} \) = \( \frac{30}{100} \) = 0.3 errors per hour

So, in an average hour, the machine will produce 6 - 0.3 = 5.7 error free parts.

The machine ran for 24 - 2 = 22 hours yesterday so you would expect that 22 x 5.7 = 125.4 error free parts were produced yesterday.