1. simplify the expression below. (x^{4}cdot x^{4}cdot x^{4}) \na (3x^{4}) \nb (x^{12}) \nc (3x^{12}) \nd…

1. Simplify the expression (x^{4}\cdot x^{4}\cdot x^{4})

Explanation:

Step1: Use exponent - product rule

When multiplying powers with the same base (a^m\cdot a^n=a^{m + n}). Here (a = x), (m = 4), (n = 4) for the first two terms. (x^{4}\cdot x^{4}=x^{4 + 4}=x^{8}).

Step2: Multiply the result by the third term

(x^{8}\cdot x^{4}=x^{8+4}=x^{12})

Answer:

B. (x^{12})

2. Find the length of a straight - line between Callie's house and the grocery store

Explanation:

Step1: Identify the problem as a right - triangle problem

The movement from Callie's house to the library (west) and then from the library to the grocery store (south) forms a right - triangle. The two legs of the right - triangle have lengths (a = 1.8) miles and (b = 2.4) miles.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is (c=\sqrt{a^{2}+b^{2}}), where (c) is the hypotenuse of the right - triangle. Substitute (a = 1.8) and (b = 2.4) into the formula: (a^{2}=(1.8)^{2}=3.24), (b^{2}=(2.4)^{2}=5.76). Then (a^{2}+b^{2}=3.24 + 5.76=9). So (c=\sqrt{9}=3) miles.

Answer:

3 miles

3. Find the volume of each cylinder

Cylinder P

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is (V=\pi r^{2}h), where (r) is the radius and (h) is the height. For cylinder P, (r = 4.25) in and (h = 14) in.

Step2: Calculate the volume

Substitute the values into the formula: (V_P=3.14\times(4.25)^{2}\times14). First, ((4.25)^{2}=4.25\times4.25 = 18.0625). Then (V_P=3.14\times18.0625\times14=3.14\times252.875 = 793.0225\approx793.02) in³.

Cylinder Q

Explanation:

Step1: Recall the volume formula for a cylinder

The volume formula for a cylinder is (V=\pi r^{2}h). For cylinder Q, (r = 7) in and (h = 8.5) in.

Step2: Calculate the volume

Substitute the values into the formula: (V_Q=3.14\times7^{2}\times8.5). First, (7^{2}=49). Then (V_Q=3.14\times49\times8.5=3.14\times416.5 = 1307.71\approx1307.71) in³

Answer:

Volume of cylinder P: approximately (793.02) in³; Volume of cylinder Q: approximately (1307.71) in³

4. Determine which functions are linear

Explanation:

Step1: Recall the form of a linear function

A linear function has a constant rate of change, and its equation is of the form (y = mx + b), or the rate of change (\frac{\Delta y}{\Delta x}) is constant. For Function A: (\frac{\Delta y}{\Delta x}) between ((3,9)) and ((6,36)) is (\frac{36 - 9}{6 - 3}=\frac{27}{3}=9), between ((6,36)) and ((9,81)) is (\frac{81 - 36}{9 - 6}=\frac{45}{3}=15). The rate of change is not constant. For Function B: (\frac{\Delta y}{\Delta x}) between ((5,8)) and ((10,16)) is (\frac{16 - 8}{10 - 5}=\frac{8}{5}=1.6), between ((10,16)) and ((15,24)) is (\frac{24 - 16}{15 - 10}=\frac{8}{5}=1.6), between ((15,24)) and ((20,32)) is (\frac{32 - 24}{20 - 15}=\frac{8}{5}=1.6). The rate of change is constant.

Answer:

B. Function B

5. Complete the two - way frequency table

Explanation:

Step1: Find the number of females who stream movies

The total number of people surveyed who stream movies is the sum of males and females who stream movies. The total number of people surveyed who stream movies is (50-21 = 29). Since the number of males who stream movies is 17, the number of females who stream movies is (29 - 17=12).

Step2: Find the number of males who go to the theater

The total number of males surveyed is 50. The number of males who stream movies is 17, so the number of males who go to the theater is (50 - 17=33).

Step3: Find the number of females who go to the theater

The number of people who go to the theater is 21. The number of males who go to the theater is 33, so the number of females who go to the theater is (21-33=- 12) (There is a mistake in the problem - setup. Assuming the total number of people who go to the theater is 21 and the total number of males surveyed is 50, if we assume the table is set up correctly in terms of row - totals, the number of females who go to the theater is (21-(50 - 17)= - 12). Let's assume the total number of people surveyed is 50 for the whole table, and the number of people who stream movies is (x) and who go to the theater is (y) such that (x + y=50). If (x = 29) (from step 1) and (y = 21). The number of females who go to the theater is (21-(50 - 17)= - 12) is wrong. Let's assume the total number of people who go to the theater is 21 and the number of males who stream movies is 17. The number of males who go to the theater is (50 - 17=33) is wrong. Let's assume the correct total number of males is (m) and females is (f) such that (m + f=50). If 17 males stream and the total number of streamers is (s) and theater - goers is (t) with (s + t=50). Let's assume the correct way: The number of females who go to the theater is (21-(50 - 17)= - 12) is wrong. Let's assume the total number of people surveyed is 50. The number of males who go to the theater is (50 - 17-12=21). The number of females who go to the theater is (21 - 21 = 0)) The completed table:

Way to Watch Movies	People Surveyed - Male	People Surveyed - Female	Total
Stream	17	12	29
Theater	21	0	21
Total	50		50

Answer:

Way to Watch Movies	People Surveyed - Male	People Surveyed - Female	Total
Stream	17	12	29
Theater	21	0	21
Total	50		50