Describe what the notation P(B|A) represents. Choose the correct answer below. A. The probability of event B or event A occurring. B. The probability of event B​ occurring, given that event A has already occurred. C. The probability of event B and event A occurring. D. The probability of event A​ occurring, given that event B has already occurred.

Answer:

Side of 22 and height of 11

Step-by-step explanation:

Let s be the side of the square base and h be the height of the tank. Since the tank volume is restricted to 5324 ft cubed we have the following equation:

[tex]V = s^2h = 5324[/tex]

[tex]h = 5324 / s^2[/tex]

As the thickness is already defined, we can minimize the weight by minimizing the surface area of the tank

Base area with open top [tex]s^2[/tex]

Side area 4sh

Total surface area [tex]A = s^2 + 4sh[/tex]

We can substitute [tex]h = 5324 / s^2[/tex]

[tex]A = s^2 + 4s\frac{5324}{s^2}[/tex]

[tex]A = s^2 + 21296/s[/tex]

To find the minimum of this function, we can take the first derivative, and set it to 0

[tex]A' = 2s - 21296/s^2 = 0[/tex]

[tex]2s = 21296/s^2[/tex]

[tex]s^3 = 10648[/tex]

[tex]s = \sqrt[3]{10648} = 22[/tex]

[tex]h = 5324 / s^2 = 5324 / 22^2 = 11[/tex]

To find the dimensions of the rectangular tank with a square base and minimum weight, we set up the volume and surface area formulas in terms of the base's side length and the tank's height, then use calculus to find the dimensions that minimize the surface area.

The problem is a typical unconstrained optimization problem in calculus. The volume of the tank is given as 5324 cubic feet. Since it's a rectangular tank with a square base, we can let x be the length of the side of the base and h be the height of the tank. Then, the volume of the tank is x^2*h = 5324.

The weight of the tank is proportional to the amount of steel used, which, in turn, is proportional to the surface area of the tank. The surface area of the tank is x^2 + 4*x*h. From the volume formula, we can express h in terms of x: h = 5324 / x^2. Substituting it to the surface area formula, we get the surface area is a function of x, such that "A = x^2 + 4x * 5324 / x^2".

To find the dimensions that minimize the weight, we find the derivative of A with respect to x and set it equal to 0. This will give the x that minimizes the surface area. Then substitute this value back into the equation h = 5324 / x^2 to get the optimal height.

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Final answer:

The total weight of books on the last book cart is 216 pounds, found by dividing the total number of books by the number of carts and then multiplying by the weight per book.

Explanation:

The question involves dividing the total number of math books evenly among book carts and then determining the total weight of books on one cart. Here's a step-by-step solution:

First, divide the total number of books, which is 288, by the number of book carts, which is 8, to find the number of books per cart: 288 \/ 8 = 36 books per cart.

Next, since each book weighs 6 pounds, multiply the number of books per cart by the weight per book to get the total weight on one book cart: 36 books x 6 pounds/book = 216 pounds.

Therefore, the total weight of books on the last book cart is 216 pounds.

To find the height of the football field at a distance of 35 feet from the side, you plug x=35 into the equation h=-0.00025x^2 + 0.04x, resulting in a height of 1.1 feet after rounding to the nearest tenth.

To find the height of the football field a distance of 35 feet from the side, we will substitute the value of x into the given quadratic equation h = -0.00025x2 + 0.04x. We can follow these steps:

Answer:

65.83

Step-by-step explanation:

The total amount Mr. And Mrs. Kingsley have to leave if Their dinner bill came to $57.24, If they wish to leave, a 15% tip is $65.83.

As it is clear from the term per cent means measuring anything per hundred. For example, you get marks per 100 marks, so you get the percentage.

Given:

The dinner bill is $57.24,

The tip percentage = 15%,

Calculate the amount of the tip as shown below,

The amount of the tip = 15% of 57.24

The amount of the tip = 15 / 100 × 57.24

The amount of the tip = 0.15 × 57.24

The amount of the tip = 8.586

Hence, total amount = tip amount + bill amount

Total amount = 57.24 + 8.586

Total amount = 65.83

Thus, the total amount is $65.83.

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Answer:it will take 0.25 years

Step-by-step explanation:

Let x represent the number of years that it will take for type A and tree B to be the same height.

Let y represent the height of type A after x years.

Let z represent the height of type B after x years.

Type A is 5 feet tall and grows at a rate of 13 inches per year. This means that

y = 5 + 13x

Type B is 3 feet tall and grows at a rate of 21 inches per year. This means that

z = 3 + 21x

To determine the number of years it will take both trees to be of the same height, we would equate equation y to z. It becomes

5 + 13x = 3 + 21x

21x - 13x = 5 - 3

8x = 2

x = 2/8 = 0.25 years

Answer: 15.07°

Step-by-step explanation:

Sin^-1 0.26= 15.07°

Therefore:

Cos15.07= 0.9656

Tan 15.07 = 0.2692

Csc15.07= 1/(Sin15.07)= 3.846

Sec15.07=1/cos15.07=1.0356

Cot15.07=1/tan15.07= 3.7147

We can use the trigonometric identity sin² θ + cos² θ = 1 to find cos θ in a first quadrant triangle where sin θ = 0.26, and get cos θ = 0.966. Then, we calculate tan θ with the formula tan θ = sin θ / cos θ, and get tan θ = 0.269 after rounding.

In trigonometry, there are three primary ratios called trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios can be defined using the sides of a right triangle, and they relate the angles of a triangle to the lengths of its sides. If we know one of these ratios in a right triangle, we can find the others.

In this case, we know that sin θ = 0.26 and θ is in Quadrant I. In the first quadrant, all the trigonometric ratios are positive.

So, firstly we find cos θ. Remember that sin² θ + cos² θ = 1, we can solve for cos θ using the calculator with the square root of (1 - sin² θ), which results in cos θ = 0.966 approximately.

Next, to find tan θ, we use the relationship tan θ = sin θ / cos θ. So, tan θ = 0.26 / 0.966 = 0.269 approximately, after rounding to the nearest hundredth.

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Answer:

The answer to your question is below

Step-by-step explanation:

a) x² = 196

   x = ±√196

  x = ± 14       or     x = -14  and x = 14

b) 9x² = 49

      x² = 49/ 9

      x = √(49/9)

      x = ±7/3

      x = ± 2.3      or  x = 2.3 and  x = -2.3

c)  4x² - 8 = 56

    4x² = 56 + 8

    4x² = 64

      x² = 64/4

      x² = 16

      x = ±√16

     x = ±4        or   x = 4 and   x = -4

d) x² = 160

   x = ± √160

Find the prime factors of 160

          160    2

            80    2

             40   2

             20   2

              10   2

               5   5

                1                  160 = 2⁵5      

 x = √2⁴10      

x =  ±4√10

e) x² = 72

    Find the prime factor of 72

    72 = 2²3²6

   x = √2²3²6

   x = ±6√6 or  

   x = ±8.5

Answer:

v(9)=2000

V(15)=8000

Step-by-step explanation:

Answer:

Step-by-step explanation:

You are given a function V(t) and asked to find V(9) and V(15). In each case, the number in parentheses replaces "t" in the function expression. After that, it is just arithmetic. (You may want to use a calculator, but it isn't really necessary.)

Given:

  V(t) = 250·2^(t/3)

Find:

  V(9) = 250·2^(9/3)

  = 250·2^3

  = 250·8

  V(9) = 2000

__

  V(15) = 250·2^(15/3)

  = 250·2^5

  = 250·32

  V(15) = 8000

_____

As you know, an exponent signifies repeated multiplication. The exponent tells you the number of times the base is a factor in the product:

  2^3 = 2·2·2 = 8 . . . . . . the exponent of 3 means 2 is a factor 3 times

Answer:

  y = -2

Step-by-step explanation:

Any asymptotes of a rational function will be described by the quotient of the numerator and denominator (excluding any remainder).

  [tex]f(x)=\dfrac{-2x}{x+1}=-2+\dfrac{2}{x+1}[/tex]

The horizontal asymptote is ...

  y = -2

Answer:

Step-by-step explanation:

The given function is

[tex]f(x)=\frac{-2x}{x+1}[/tex]

An asymptote refers to a restriction in the domain or range set of the function. This happens to rational functions, because there's a scenario where the function is undetermined: when its denominator is zero.

So, in this case, the value that makes the denominator zero is

[tex]x+1=0\\x=-1[/tex]

That means, we need to restrict the domain for [tex]x=-1[/tex]. But this is a vertical asymptote.

To find horizontal asymptotes, we need to find the restrictions for the range of the function, we do that isolating [tex]y[/tex].

[tex]y=\frac{-2x}{x+1}\\yx+y=-2x\\yx+2x=-y\\x(y+2)=-y\\x=-\frac{y}{y+2}[/tex]

So, if we analyse the denominator

[tex]y+2=0\\y=-2[/tex]

Therefore, the restriction is [tex]y=-2[/tex]. In other words, the vertical asymptote is the first choice.

There are 659 blocks

Solution:

The number in ones place is 9

Let's denote number in the tens place with x

The number in the hundreds place is one more then the tens place

Therefore,

The number in hundreds place is 1 + x

Those two numbers equal 11

Which means tens place number and hundreds place number sums up to 11

tens place + hundreds place = 11

x + 1 + x = 11

2x = 11 - 1

2x = 10

x = 5

Thus number in tens place = x = 5

Number in hundreds place = 1 + x = 1 + 5 = 6

The number is represented as:

Number = (Number in hundreds place)(number in tens place)(number in ones place)

Number = 659

Thus 659 blocks are there

Answer:

The coordinate of [tex]Z=7.3[/tex].

Step-by-step explanation:

To determine:

What is the coordinate of Z?

Fetching Information and Solution Steps:

The distance [tex]XY[/tex] = [tex]3.8-1.3 = 2.5[/tex]

As

Y is [tex]\frac{5}{12}[/tex] of the way from [tex]X[/tex] to [tex]Z[/tex]

So,

[tex]YZ[/tex] is [tex]\frac{7}{5}[/tex] as big as [tex]XY[/tex]

Therefore,

[tex]Z = 3.8 + (\frac{7}{5} )(2.5)[/tex]

[tex]Z=3.8+\frac{17.5}{5}[/tex]

[tex]Z=7.3[/tex]

Therefore, the coordinate of [tex]Z=7.3[/tex].

Keywords: coordinate, point, length, ratio

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The coordinate of Z is (4.43, 11.57)

The coordinate of X is (1, 3)

The coordinate of Y is (3, 8)

Let the coordinate of z be (x₂, y₂)

The ratio of division is 5:7

m = 5, and n = 7

Let the coordinate of y be (a, b)

a = 3, b = 8

[tex]a=\frac{mx_1+nx_2}{m+n}\\\\3= \frac{5(1)+7x_2}{5+7}\\\\3(12)=5+7x_2\\\\7x_2=36-5\\\\7x_2=31\\\\x_2=4.43[/tex]

[tex]ab=\frac{my_1+ny_2}{m+n}\\\\8= \frac{5(3)+7y_2}{5+7}\\\\8(12)=15+7y_2\\\\7y_2=96-15\\\\7y_2=81\\\\y_2=11.57[/tex]

The coordinate of Z is (4.43, 11.57)

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Answer:a) C = 2πr

b) 37.68feet

c) r = C/2π

d) 5.89feet

Step-by-step explanation:

The answers are a) 2πr, b) 37.68 ft, c) r = C/2π and d) 5.8 ft

The circumference is the distance around the edge of a circle.

Given that, a circle whose size can vary. The circumference of the circle is always 2π times as large as its radius and r represent the radius of the circle (in feet) and let C represent the circumference of the circle (in feet).

a) C = 2πr

b) C = 2*3.14*6 = 37.68ft

c) C = 2πr, r = C/2π

d) r = 37/2*3.14 = 5.8 ft

Hence, The answers are a) 2πr, b) 37.68 ft, c) r = C/2π and d) 5.8 ft

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Answer:

2.

Step-by-step explanation:

Healthy baby.

Answer: a) Length of wire to replace old one=9ft

b) Length of wire=3.09ft

Leftover wire=0.91ft

Step-by-step explanation: from the right angle triangle, Tan&=20/15

&=53.13°

Length of wire,x=cos 31.13=x/15

15cos31.13

X=9

b) Tan&=10/6

Tan^-110/6

&=59.05

Length= 6cos59.04

X=3.09

4-3.09=.91= leftover wire

Using the Pythagorean theorem, the length of the guy wire needed for a 20-foot flagpole with a 15-foot distant anchor point is 25 feet, and for a 10-foot flagpole with a 6-foot distant anchor point, a 12-foot guy wire is needed, leaving approximately 4 inches of wire leftover.

The question involves applying the Pythagorean theorem to determine the lengths of guy wires needed to support flagpoles. For the first scenario, a 20-foot tall flagpole with a 15-foot distant anchor point, the guy wire's length can be found using the equation a^2 + b^2 = c^2, where 'c' is the hypotenuse (the guy wire), 'a' is one leg (the height of the flagpole which is 20 feet), and 'b' is the other leg (the distance from the base of the pole to the anchor point which is 15 feet). After calculation, the guy wire's length is found to be 25 feet.

For the second scenario with a 10-foot tall flagpole and a 6-foot distant anchor point, using the same theorem gives us the equation 10^2 + 6^2 = c^2. Estimating using perfect squares, we get c as the square root of 136, which is approximately 11.66 feet. Since guy wires can only be purchased in whole foot lengths, we would purchase a 12-foot guy wire, leaving approximately 0.34 feet (about 4 inches) of wire leftover.

Answer:

a). 170 miles

b). Planes will be 500 miles apart after 1.1 hours.

Step-by-step explanation:

a). Two planes heading towards North and South when passed each other were 80 miles apart.

Distance between these planes after 20 minutes or [tex]\frac{1}{3}[/tex] hours.

From the figure attached,

In right angle triangle DEC,

DE = 80 miles

BE = (Speed × duration) = [tex]300\times \frac{1}{3}=100[/tex] miles

Similarly, BC = [tex]150\times \frac{1}{3}=50[/tex] miles

By Pythagoras theorem,

DC² = EC² + DE²

       = (EB + BC)² + DE²

       = (100 + 50)² + (80)²

       = 28900

DC = √28900 = 170 miles

b). Now we have to evaluate the duration after which distance between the planes is 500 miles.

Let after t hours planes will be 500 miles apart.

Then EB = 300t

BC = 150t

Therefore, EC = EB + BC = 450t

It's given that DC = 500 miles

By Pythagoras theorem again,

DC² = EC²+ DE²

(500)²=  (450t)²+ (80)²

250000 = 202500t² + 6400

2500 = 2025t² + 64

2025t² = 2436

t² = 1.20297

t = 1.097 hours ≈ 1.1 hours

Therefore, both the planes will be 500 miles apart after 1.1 hours.

Answer: Probability that at least two of the triplets will win a medal is19/24

Step-by-step explanation:

Total number of cmpetitors=9

Total number of selecting 3 winners out of 9 competitors=9C3=84

Number of ways of selecting 2 out of 3 triplets=3C2

Number of selecting 1 out of 6 others =6C1

Number of selecting 2 out of 3 triplet and o1 out of 6 others= 3C2 ×6C1=18

Number of selecting 3 out of 3 triplets=3C3=1

Total ways of selecting less than 2 or 2 triplets= 18+1=19

Probability that at least 2 of the triplets will win is 19/84

Answer:

(x) free throw = 8

(y) 2-point = 6

(z) 3-point = 2

x+(y*2)+(z*3)

8+(6*2)+(2*3)

8+12+6

=26

Answer:

-5/6

Step-by-step explanation:

covert the mixed number into a improper fraction

4/3 + (- 13/6) =

Find LCD which is 6 so

[tex]\frac{4}{3} * 2[/tex]

8/6 + (-13/6) =

Which is -5/6

The answer is

[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]

=======================================================

Work Shown:

[tex](-3+\sqrt{-2})(3-\sqrt{2})[/tex]

[tex](-3+\sqrt{-1*2})(3-\sqrt{2})[/tex]

[tex](-3+\sqrt{-1}*\sqrt{2})(3-\sqrt{2})[/tex]

[tex](-3+i\sqrt{2})(3-\sqrt{2})[/tex]

[tex]x(3-\sqrt{2})[/tex] let [tex]x=-3+i\sqrt{2}[/tex]

[tex]3x-x*\sqrt{2}[/tex] distribute

[tex]3(x)-\sqrt{2}(x)[/tex]

[tex]3(-3+i\sqrt{2})-\sqrt{2}(-3+i\sqrt{2})[/tex]

[tex]-9+3i\sqrt{2}+3\sqrt{2}-i*\sqrt{2}*\sqrt{2}[/tex]

[tex]-9+3i\sqrt{2}+3\sqrt{2}-2i[/tex]

[tex]-9+3\sqrt{2}+3i\sqrt{2}-2i[/tex]

[tex](-9+3\sqrt{2})+(3\sqrt{2}-2)i[/tex]

The use of parenthesis in the last step is to help separate out the terms.

The last expression shown above is in the form a+bi where

[tex]a=-9+3\sqrt{2}[/tex]

[tex]b=3\sqrt{2}-2[/tex]

Answer:

y=22000x+40000

Answer: the cost of one adult ticket is $8

the cost of one kid ticket is $4

Step-by-step explanation:

Let x represent the cost of one adult ticket.

Let y represent the cost of one kid ticket.

They sold 64 adult tickets (x) and 132 kid tickets (y) and made $1040. This means that

64x + 132y = 1040 - - - - - - - - - - -1

An adult ticket is double the cost of a kid's ticket. This means that

x = 2y

Substituting x = 2y into equation 1, it becomes

64 × 2y + 132y = 1040

128y + 132y = 1040

260y = 1040

y = 1040/260 = 4

x = 2y = 2 × 4

x = 8

Answer:

A= 1132.8 m^2

Step-by-step explanation:

l=19.1m

b=18m

h=12m

s=15m

The formula to find surface area of the triangular prism is:

A = bh + 2ls + lb

A = (18×12) + 2(19.1×15) + (19.1×18)

A = 216 + 573 + 343.8

A= 1132.8 m^2

Answer: surface Area of triangular prism = 502.5m^2

Step-by-step explanation:

Formular for surface area of a prism is given as: SA=bh+(S1 +S2+S3)H

Given base =18m, length=19.1, h=12, w=15

S1=1/2bh =1/2×18×12 =108m

S2=108m . This is because the dimensions of the two ends of the triangular prism.

S3=lw=19.1×15=286.5m

Surface area=108+108+286.5=502.5m^2

Answer:

August D is the fastest rapper. XD

Step-by-step explanation:

Answer:

Based on this info I would say that Mark Lee does because he is at 90.0 sps.

Step-by-step explanation:

Question is Incomplete; Complete question is given below;

Chase purchased a sweatshirt from the clearance rack. The price of the sweatshirt after the discount is represented by the expression ​0.75x, where ​x  represents the original price of the sweatshirt.

Which expression also represents the discounted price of the sweatshirt?

A  (0.75​ −​ 0.25)x  

B  ​(0.75​ +​ 0.25)x  

C  x​ –​ 0.25x  

D  ​x​ +​ 0.25x

Answer:

C . [tex]x-0.25x[/tex]

Step-by-step explanation:

Given:

Expression representing price of the sweatshirt after the discount = [tex]0.75x[/tex]

[tex]x[/tex] ⇒ original price of sweatshirt

We need to find the expression which also represents the discounted price of the sweatshirt.

Solution:

Form the given expression we can see that after discount we are paying only 75% of the original amount of sweatshirt.

So we can say that;

The discount price was 25% of the original price i.e [tex]0.25x[/tex]

So now we can say that;

Price after discount is equal to difference of original price and discounted price.

framing in equation form we get;

Price after discount = [tex]x-0.25x[/tex]

Hence the equivalent expression for the given discounted price of sweatshirt is  [tex]x-0.25x[/tex].

Answer:

Step-by-step explanation:

Answer:

His diagonal brace need to be 15.55 feet long in the kitchen.

Step-by-step explanation:

Given : The wall of the kitchen is in the shape of a square.

Area of the wall = 121 sq ft

Let us assume the each side of  wall = a ft

Now, AREA OF THE SQUARE = Side x Side

⇒   a x  a = 121 sq units

[tex]\implies a ^2 = 121\\\implies a = 11[/tex]

So, each side of the  kitchen wall is 11 ft.

Now,as  we know in a square : EACH ANGLE IN THE  SQUIRE IS A RIGHT ANGLE.

So, by Pythagoras Theorem, Diagonal D is:

[tex]D^2 = a^2 + a^2\\\implies D^2 = 11^2 + 11^2 = 121 + 121\\\implies D^2 = 242\\\implies D = 15.55[/tex]

Hence,  his diagonal brace need to be 15.55 ft long in the kitchen.

Answer:

four numbers are 1, 3, 5, 7

Step-by-step explanation:

The sum of four numbers in arithmetic progression is 16

a, a+d, a+2d, a+3d are the four arithmetic series

sum of 4 numbers are

[tex]a+a+d+a+2d+a+3d=4a+6d[/tex]

[tex]4a+6d= 16[/tex]

divide both sides by 2

[tex]2a+3d=8[/tex]

[tex]3d= 8-2a[/tex]

The square of the last number is the square of the first number plus 48.

[tex](a+3d)^2=a^2+48[/tex]

solve for  a  and d

[tex](a+3d)^2=a^2+48\\(a+8-2a)^2=a^2+48\\(8-a)^2=a^2+48\\a^2-16a+64=a^2+48\\-16a=48-64\\-16a=-16\\a=1[/tex]

Now find out 'd'

[tex]3d=8-2a\\3d=8-2\\3d=6\\a=2[/tex]

a, a+d, a+2d, a+3d

four numbers are 1, 3, 5, 7

Answer:

[tex]4x+9+4x^2[/tex]

Step-by-step explanation:

The given functions are:

[tex]f(x)=4x+9[/tex] and [tex]g(x)=4x^2[/tex]

We want to find [tex](f+g)(x)[/tex].

Recall that:

[tex](f+g)(x)=f(x)+g(x)[/tex]

We substitute the functions to get:

[tex](f+g)(x)=4x+9+4x^2[/tex]

We now write in standard form to obtain:

[tex](f+g)(x)=4x^2+4x+9[/tex]

If the quantity demanded, q, of a product when the price is p dollars is given by the equation p = 462 − 5q, the equilibrium price and quantity are 132 dollars and 66.

An equation is a combination of expressions in a meaningful manner in terms of variables.

The given equation between the quantity demanded, q, of a product and the price, p dollars, is as follows:

p = 462 - 5q ...(1)

The quantity supplied is given by the equation as follows:

p = 2q ...(2)

Substitute p = 2q in the equation (1),

2q = 462 - 5q

2q + 5q = 462

7q = 462

[tex]q = \dfrac{462}{7}[/tex]

q = 66

The equilibrium quantity is 66.

Put q = 66 in equation (2),

p = 2 × 66

p = 132

The equilibrium price is 132 dollars.

Thus, the equilibrium price and quantity are 132 dollars and 66.

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The equilibrium quantity is 66 units, and the equilibrium price is $168.

To find out the equilibrium price and quantity, we must make the quantity demanded equal to the quantity supplied. This essentially means that we need to solve the system of equations defined by the two product price functions therefore set 462 - 5q = 2q.

The next step is grouping the 'q' terms on one side and the constants on the other which gives us: 7q = 462. Dividing by 7 on both sides, we get q = 66, which is the equilibrium quantity.

Finally, by substituting q = 66 into either of the original equations (but be consistent, for this case I'll use the first equation: p = 462 - 5q), we find out that the equilibrium price is p = 462 - 5*66 = $168.

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Explanation

Lets use congruence

For the square of an integer lets use congruence module 3

Thus, the square of an integer has the form 3k (the congruence is 0) or 3k+1 (the congruence is 1).

For the cube, lets use congruence module 9

With the argument given above, we obtain that the congruence module 9 of the cube of a number is always 0,1 ir 8, thus, the cube of an integer has the form 9k, 9k+1 or 9k+8.

As for the fourth power, we take congruence module 5:

In any case, the congruence of the fourth power module 5 is either 0 or 1, as a result, the fourth power has the form 5k or 5k+1.

Final answer:

Using the Division Algorithm, we show that the square of an integer is in the form of 3k or 3k+1, the cube is in the form of 9k, 9k+1, or 9k+8, and the fourth power is in the form of 5k or 5k+1.

Explanation:

The Division Algorithm states that if we divide any integer n by a positive integer d, there are unique integers q and r such that n = dq + r and 0 ≤ r < d. Using this and thinking about integers as being of the form 3q, 3q + 1, or 3q + 2 we can prove the following statements:

(a) The square of any integer:

Any integer can be written as 3k, 3k + 1, or 3k + 2. Squaring these forms:

In each case, the square of an integer is either of the form 3k or 3k + 1.

(b) The cube of any integer:

Cubing the three forms:

The cube is of the form 9k, 9k + 1, or 9k + 8.

(c) The fourth power of any integer:

For 3k and 3k + 1, the fourth powers are:

For 3k + 2, because 2⁴ = 16 is already a multiple of 5, (3k + 2)⁴ will also have the form 5m or 5m + 1. Therefore, the fourth power of an integer is either of the form 5k or 5k + 1.