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:

2z⁴ x z⁷
(2 x 1)z^{(4 + 7)}
2z¹¹

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

3\( \sqrt{50} \) - 3\( \sqrt{2} \)
3\( \sqrt{25 \times 2} \) - 3\( \sqrt{2} \)
3\( \sqrt{5^2 \times 2} \) - 3\( \sqrt{2} \)
(3)(5)\( \sqrt{2} \) - 3\( \sqrt{2} \)
15\( \sqrt{2} \) - 3\( \sqrt{2} \)

Now that the radicands are identical, you can subtract them:

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

\( \frac{7}{100} \) x 10 = \( \frac{7 \times 10}{100} \) = \( \frac{70}{100} \) = 0.7 errors per hour

So, in an average hour, the machine will produce 10 - 0.7 = 9.3 error free parts.

The machine ran for 24 - 3 = 21 hours yesterday so you would expect that 21 x 9.3 = 195.3 error free parts were produced yesterday.

If the tiger has consumed 64 pounds of food in 8 days that's \( \frac{64}{8} \) = 8 pounds of food per day. The tiger needs to consume 96 - 64 = 32 more pounds of food to reach 96 pounds total. At 8 pounds of food per day that's \( \frac{32}{8} \) = 4 more days.

The distributive property for multiplication helps in solving expressions like a(b + c). It specifies that the result of multiplying one number by the sum or difference of two numbers can be obtained by multiplying each number individually and then totaling the results: a(b + c) = ab + ac. For example, 4(10-5) = (4 x 10) - (4 x 5) = 40 - 20 = 20.