Question

1. which of the following is equal to \\((xy^2)^3\\)?

a) \\(x^3y^5\\)
b) \\(x^3y^6\\)
c) \\(x^3y^2\\)
d) \\((x^3y)^2\\)

1. doing volunteer cleanup at a park, sravani can collect three bags of trash in an hour, whereas liam can collect two bags in 40 minutes. working together, how long would it take them to collect 72 bags total?

a) 6 hours
b) 8 hours
c) 10 hours
d) 12 hours

1. what is the circumference of a circle with an area of \\(16\pi \\, \text{m}^2\\)?

a) \\(8\pi \\, \text{m}\\)
b) \\(12\pi \\, \text{m}\\)
c) \\(16\pi \\, \text{m}\\)
d) \\(64\pi \\, \text{m}\\)

1. you are starting a new bakery that sells the following types of items: muffins, scones, croissants, and doughnuts. you decide to sell a new combo meal consisting of two items of two different types (e.g. one muffin and one scone, or one scone and one croissant, but not two scones). how many different combinations of item types could you make?

a) 4
b) 6
c) 8
d) 10

1. which of the following expressions has the greatest value?

a) \\(20 - 5\\)
b) \\(|-5 - 20|\\)
c) \\(|-5| - |20|\\)
d) \\(|20 - 5|\\)

Explanation:

Question 10

Step1: Apply exponent rule \((ab)^n = a^n b^n\)

\((xy^2)^3 = x^3 (y^2)^3\)

Step2: Apply exponent rule \((a^m)^n = a^{mn}\)

\(x^3 (y^2)^3 = x^3 y^{2\times3} = x^3 y^6\)

Step1: Find Liam's rate

Liam collects 2 bags in 40 minutes. Convert 40 minutes to hours: \(40\) minutes \(=\frac{40}{60}=\frac{2}{3}\) hours. Rate \(=\frac{2}{\frac{2}{3}} = 3\) bags per hour.

Step2: Find combined rate

Sravani's rate: 3 bags per hour. Combined rate \(= 3 + 3 = 6\) bags per hour.

Step3: Find time for 72 bags

Time \(=\frac{72}{6}=12\) hours? Wait, no, wait. Wait Sravani: 3 bags per hour. Liam: 2 bags in 40 minutes, so in 60 minutes (1 hour), he collects \(\frac{2}{\frac{40}{60}} = 3\) bags? Wait 40 minutes is \(\frac{2}{3}\) hour, so rate is \(2\div\frac{2}{3}= 3\) bags per hour. So combined rate is \(3 + 3 = 6\) bags per hour. Then time for 72 bags is \(72\div6 = 12\)? But wait the options have 8? Wait no, maybe I miscalculated Liam's rate. Wait 40 minutes is \(\frac{2}{3}\) hour, so 2 bags in \(\frac{2}{3}\) hour, so rate is \(2\div\frac{2}{3}= 3\) bags per hour. Sravani: 3 bags per hour. So together 6 bags per hour. 72 bags: \(72\div6 = 12\) hours. But option D is 12. Wait maybe I made a mistake. Wait let's check again. Sravani: 3 bags per hour. Liam: 2 bags in 40 minutes. 40 minutes is \(\frac{2}{3}\) hour. So rate is \(2\) bags / \(\frac{2}{3}\) hour \(= 3\) bags per hour. So combined rate 3 + 3 = 6. Time = 72 / 6 = 12. So answer D.

Step1: Find radius from area

Area of circle \(A = \pi r^2\). Given \(A = 16\pi\), so \(\pi r^2 = 16\pi\). Divide both sides by \(\pi\): \(r^2 = 16\), so \(r = 4\) m.

Step2: Find circumference

Circumference \(C = 2\pi r\). Substitute \(r = 4\): \(C = 2\pi\times4 = 8\pi\) m.

Answer:

B) \(x^3y^6\)

Question 11