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:

7z² x 7z⁶
(7 x 7)z^{(2 + 6)}
49z⁸

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 45% the radius (and, consequently, the total area) increases by \( \frac{45\text{%}}{2} \) = 22\(\frac{1}{2}\)%

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 7 = \( \frac{9 \times 7}{100} \) = \( \frac{63}{100} \) = 0.63 errors per hour

So, in an average hour, the machine will produce 7 - 0.63 = 6.37 error free parts.

The machine ran for 24 - 3 = 21 hours yesterday so you would expect that 21 x 6.37 = 133.8 error free parts were produced yesterday.

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:

-3y⁶ + 1y⁶
(-3 + 1)y⁶
-2y⁶

First, solve for \( \left|y - 3\right| \):

\( \left|y - 3\right| \) + 5 = -6
\( \left|y - 3\right| \) = -6 - 5
\( \left|y - 3\right| \) = -11

The value inside the absolute value brackets can be either positive or negative so (y - 3) must equal - 11 or --11 for \( \left|y - 3\right| \) to equal -11:

So, y = 14 or y = -8.