Mathematics College

There are six different sixth roots of 64. That is, there are six complex numbers that solve x^6=64

Please help

Answers

Answer 1
There are two possible solutions:

1)
When we know that:

[tex]\sqrt[n]{z}=\sqrt[n]{|z|}\Big(\cos\frac{\phi+2k\pi}{n}+i\sin\frac{\phi+2k\pi}{n}\Big)\qquad\text{for}\quad k=0,1,\ldots,n-1[/tex]

when [tex]z=|z|(\cos\phi+i\sin\phi)[/tex]

then:

[tex]x^6=64\quad|\sqrt[6]{(\ldots)}\\\\x=\sqrt[6]{64}[/tex]

For [tex]x=64[/tex] we have:

[tex]64=|64|(\cos0+i\sin0)\quad\Rightarrow\quad \phi=0[/tex]

and:

[tex]\sqrt[6]{64}=\sqrt[6]{64}\Big(\cos\frac{0+2k\pi}{6}+i\sin\frac{0+2k\pi}{6}\Big)\qquad\text{for}\quad k=0,1,\ldots,5\\\\\\ \sqrt[6]{64}=2\Big(\cos\frac{k\pi}{3}+i\sin\frac{k\pi}{3}\Big)\qquad\text{for}\quad k=0,1,\ldots,5 [/tex]

k = 0

[tex]2\Big(\cos\frac{0}{3}+i\sin\frac{0}{3}\Big)=2(1+0i)=2\cdot1=\boxed{2}[/tex]

k = 1

[tex]2\Big(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\Big)=2\Big(\frac{1}{2}+i\frac{\sqrt{3}}{2}\Big)=\boxed{1+i\sqrt{3}}[/tex]

k = 2

[tex]2\Big(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\Big)=2\Big(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\Big)=\boxed{-1+i\sqrt{3}}[/tex]

k = 3

[tex]2\Big(\cos\frac{3\pi}{3}+i\sin\frac{3\pi}{3}\Big)=2\Big(\cos\pi+i\sin\pi\Big)=2(-1+0i)=\boxed{-2}[/tex]

k = 4

[tex]2\Big(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\Big)=2\Big(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\Big)=\boxed{-1-i\sqrt{3}}[/tex]

k = 5

[tex]2\Big(\cos\frac{5\pi}{3}+i\sin\frac{5\pi}{3}\Big)=2\Big(\frac{1}{2}-i\frac{\sqrt{3}}{2}\Big)=\boxed{1-i\sqrt{3}}[/tex]

So the answer is:

[tex]x=\{2,\,1+i\sqrt{3},\,-1+i\sqrt{3},\,-2,\,-1-i\sqrt{3},\,1-i\sqrt{3}\}[/tex]

2)
We don't know method (1). If so, we could use following identities:

[tex](1)\quad a^2-b^2=(a+b)(a-b)\\\\(2)\quad a^3-b^3=(a-b)(a^2+ab+b^2)\\\\(3)\quad a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]

There will be:

[tex]x^6=64\\\\x^6-64=0\\\\(x^3)^2-8^2=0 \qquad\text{from (1)}\\\\(x^3+8)(x^3-8)=0\\\\(x^3+2^3)(x^3-2^3)=0\qquad\text{from (2) and (3)}\\\\ (x+2)(x^2-2x+4)(x-2)(x^2+2x+4)=0\qquad(\star)[/tex]

Now, we complete the square for:

[tex]x^2-2x+4=x^2-2x+1+3=(x^2-2x+1)+3=(x-1)^2+3=\\\\=(x-1)^2+(\sqrt{3})^2=(x-1)^2-(-1)(\sqrt{3})^2=(x-1)^2-i^2(\sqrt{3})^2=\\\\=(x-1)^2-(i\sqrt{3})^2=\text{from (1)}=\boxed{(x-1-i\sqrt{3})(x-1+i\sqrt{3})}[/tex]

and for:

[tex]x^2+2x+4=x^2-2x+1+3=(x^2+2x+1)+3=(x+1)^2+3=\\\\=(x+1)^2+(\sqrt{3})^2=(x+1)^2-(-1)(\sqrt{3})^2=(x+1)^2-i^2(\sqrt{3})^2=\\\\=(x+1)^2-(i\sqrt{3})^2=\text{from (1)}=\boxed{(x+1-i\sqrt{3})(x+1+i\sqrt{3})}[/tex]

When we return to [tex](\star)[/tex]:

[tex](x+2)(x^2-2x+4)(x-2)(x^2+2x+4)=0\\\\(x+2)(x-1-i\sqrt{3})(x-1+i\sqrt{3})(x-2)(x+1-i\sqrt{3})(x+1+i\sqrt{3})=\\=0[/tex]

And we have answer:

[tex]x=\{-2,\,1+i\sqrt{3},\,1-i\sqrt{3},\,2,\,-1+i\sqrt{3},\,-1-i\sqrt{3}\} [/tex]
Answer 2

To find the six sixth roots of 64, you can use polar form and De Moivre's Theorem to calculate the complex numbers. The roots are 2, 1+i√3, -1+i√3, -2, -1-i√3, and 1-i√3.

[tex]X^6 = 64[/tex]

The sixth roots of 64 are complex numbers and can be found by writing 64 in polar form and applying De Moivre's Theorem. Let's find the six roots:

First root: 2(cos(0) + i sin(0)) = 2

Second root: 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3

Third root: 2(cos(2π/3) + i sin(2π/3)) = 2(-1/2 + i√3/2) = -1 + i√3

Fourth root: -2

Fifth root: -2(cos(-π/3) + i sin(-π/3)) = -2(1/2 - i√3/2) = -1 - i√3

Sixth root: -2(cos(-2π/3) + i sin(-2π/3)) = -2(-1/2 - i√3/2) = 1 - i√3

Related Questions

Find the quotient of 5/68

Answers

Use a calculator 5/68 = 5 ÷ 68 = ______
The answer for 5 divided by 68 is 0.073529411764706

It takes 54 minutes for 4 people to paint 6 walls. How many minutes does it take 6 people to paint 7 walls?

Answers

it takes 35 mins for 6 people to paint 7 walls if you would like this answer described to you let me know please 

Answer:

It will take 42 minutes.

Step-by-step explanation:

In order to solve this we first have to calulate how long will it take one person to paint one wall:

54/6=9minutes

That is between 4 people it will take 9 minutes to paint a wall, one person alone supposing they all work at the same rate will take 36 minutes.

Now we just multiply that by the number of walls to be painted:

36*7=252

And we divide that by the number of people painting:

252/6=42

So it will take 42 minutes to paint 7 walls with 6 people painting.

SIMPLIFY the expression! Please!

Answers

1+3^2 = 1+9 = 10

10/5 = 2

2-6+2 = -4+2

 answer is -2

Your answer is -2 hope this helps!

The area of a circle is 99 in.² what’s the radius

Answers

The radius is ≈ 5.61in. I hope this helps love! :)

What is the common ratio for this geometric sequence?

Answers

Answer is c 3 because it is multiplied 3 everytime

Answer:

Common ratio for the sequence is 3

Step-by-step explanation:

Given sequence is

5, 15, 45, 135,.....

To find out common ratio, divide the second number by the first number

[tex]\frac{15}{5} =3[/tex]

[tex]\frac{45}{15} =3[/tex]

[tex]\frac{135}{45} =3[/tex]

we got 3 that is our common ratio 3

Common ratio for the sequence is 3

caryssa makes bracelets to sell at craft fairs. The supplies she needed to start her business cost $45. In addition, the supplies for each bracelet cost her $3. She sells her bracelets for $5. What is the minimum number of braclets she must sell to make a profit?

Answers

at least 10 bracelets


23 because she makes $2 profit by selling bracelets. So she needs to get up to $46 in order to gain a profit. 2 x 23 bracelets = 46$

The team manager mixes 0.6 kg of Gatorade powder with water in the 5-gallon water coolers for the football team. Gatorade donates new 7-gallon water jugs. How much powder should the manager use in a new 7-gallon cooler to maintain the same Gatorade mixture?

Answers

Since it's 0.6 kg for 5 gallons, we'd want to find the unit rate aka the amount of kg's per gallon. To find that, we have to see how many times 5 goes into 0.6, which is 0.6/5=0.12 for 1 gallon. Multiply that by 7 to get 0.84 for 7 gallons!

You are at a Carnival .One of the Carnival games ask you to pick a door and then pick a curtain behind the door. There are 3 doors and 4 curtains behind each door.how many choices are possible for the player ?????

Answers

Number of possibilities = number of doors x number of curtains

3x4=12

Therefore, there are 12 possibilities.

There are 12 possible choices for the player.

What is combination?

"It is a way of selecting items from a collection "

What is formula of combination?

"[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]"

What is n factorial?

n! = n × (n - 1) × . . . × 2 × 1

For given question,

There are 3 doors and 4 curtains behind each door.

The number of possible ways of selecting a door from the 3 doors.

Using combination formula,

[tex]^3C_1=\frac{3!}{1!(3-1)!}\\\\ ^3C_1=\frac{3\times 2!}{2!}\\\\ ^3C_1=3[/tex]

The number of possible ways of selecting a curtain from the 4 curtains.

Using combination formula,

[tex]^4C_1=\frac{4!}{1!(4-1)!}\\\\ ^4C_1=\frac{4\times 3!}{3!}\\\\ ^3C_1=4[/tex]

So, the number of choices for the player to pick a door and then pick a curtain behind the door :

[tex]^3C_1\times ^4C_1\\\\=3\times 4\\\\=12[/tex]

Therefore, there are 12 possible choices for the player.

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-0.7 repeating as a decimal in simplest form

Answers

hello : 
- 0.7 = - 7/10

The floor of a one-story building is 14 feet longer than it is wide. The building has 1715 square feet of floor space. Find the length l and width w of the floor.

Answers

        w=widthlength=w+14A=lww(w+14)= 1395w(w+14)-1395=0w2+14w-1395=01395=3x3x5x31 and (3x3x5)-31=45-31=14(w+45)(w-31)=0w-31=0w=31w+14=31+14=45length=45 feetwidth=31 feet

Final answer:

To find the floor dimensions of a building that is 14 feet longer than it is wide with an area of 1715 sq ft, we solve the quadratic equation w * (w + 14) = 1715, resulting in w=100 feet and length l=114 feet.

Explanation:

The student is asked to find the length (l) and width (w) of a one-story building floor which is 14 feet longer than it is wide, with a total area of 1715 square feet. To solve this, we set up an equation where w represents the width and w + 14 feet represents the length. Since the area of a rectangle is length times width, the equation is w * (w + 14) = 1715.

After solving this quadratic equation, we find that w=100 feet which makes the length l = w + 14 = 114 feet. Therefore, the actual dimensions of the building are 114 ft by 100 feet.

Please help!!
A road trip of 400 miles require a total of 8 hours to drive. For the first portion of the trip, an average speed of 63 miles per hour was maintained; on the last portion of the trip, the average speed was only 443 miles per hour due to highway construction project. How much time, in hours, to the nearest tenth of an hour, was spent driving at each of the two speeds, and what was the total time for the entire trip?

Answers

2.8 hours at 63 mph 5.2 hours at 43 mph Total time spent driving = 8.0 hours. I'm assuming based upon context that your value of 443 miles per hour is a typo and the correct value is 43 miles per hour. The solution provided is for the two speeds of 63 mph and 43 mph. First make an expression detailing what you know. X = duration spent at 63 mph (8 - X) = duration spent at 43 mph 63X + 43(8-X) = 400 Solve for X 63X + 43(8-X) = 400 63X + 344 - 43X = 400 63X - 43X = 56 20X = 56 X = 2.8 So you spent 2.8 hours at 63 mph and 8-2.8 = 5.2 hours at 43 mph

You and your brother are readibg the same novel you wnat to get ahead of him in the book so you decide to read 30 minutes longer than your brother reads how many minutes will you read if your brother reads for 15 minutes

Answers

45 minute 30+15= 45 easy

Answer:

45 minute 30+15= 45 easy

Step-by-step explanation:

what is 2x+square root x+1=8 solve for x

Answers

[tex]\bf 2x+\sqrt{x+1}=8\implies \sqrt{x+1}=8-2x\quad \leftarrow \textit{now we square both sides} \\\\\\ (\sqrt{x+1})^2=(8-2x)^2\implies x+1=8^2-32x+(2x)^2 \\\\\\ x+1=64-32x+2^2x^2\implies x+1=64-32x+4x^2 \\\\\\ 0=4x^2-32x-x+64-1\implies 0=4x^2-33x+63 \\\\\\ 0=(4x-21)(x-3)\implies \begin{cases} 0=4x-21\\ 21=4x\\\\ \boxed{\frac{21}{4}=x}\\ --------\\ 0=x-3\\ \boxed{3=x} \end{cases}[/tex]
Hello,

On my keybord i have no touch "square root (√)" but "V" yes .

[tex]2x+ \sqrt{x+1} =8\\ \sqrt{x+1} =8-2x\\ x+1=(8-2x)^2\\ x+1=64-32x+4x^2\\ 4x^2-33x+63=0\\ x= \frac{21}{4} \ or\ x=3\\ [/tex]

Verification:
if x=3 then 2*3+√(3+1)=6+2=8

if x=21/4 then 13≠8 is to exclude.

The distance d that a certain particle moves may be calculated from the expression d = at + bt2 where a and b are constants; and t is the elapsed time. what must the dimensions of the quantities a and b be, respectively?

Answers

Guven that the distance d that a certain particle moves may be calculated from the expression [tex]d = at + bt^2[/tex] where a and b are constants; and t is the elapsed time.

Distance is a length and hence the dimension of distance is L.

Now, [tex]at[/tex] and [tex]bt^2[/tex] also will have the dimension of L.

Time has a dimension of T.

For [tex]at[/tex], let the dimension of [tex]a[/tex] be [tex]X[/tex], then
[tex]XT=L \\ \\ \Rightarrow X=\frac{L}{T}[/tex]

For [tex]bt^2[/tex], let the dimension of [tex]b[/tex] be [tex]Y[/tex], then
[tex]YT^2=L \\ \\ \Rightarrow Y= \frac{L}{T^2} [/tex]

Therefore, the dimension of [tex]a[/tex] is [tex]\frac{L}{T}[/tex] while the dimension of [tex]b[/tex] is [tex]\frac{L}{T^2}[/tex].

The total cost of the fatigato family's two cars was 71,482. The cost of one car was 38,295. Write an equation using a variable to represent the cost of the familys other car.

Answers

The total cost of the cars owned by the Fatigato family can be calculated by adding up the individual cost of the cars. We are given that the family has two cars. The cost of one car is also given. To be able to answer this item, we let x be the missing cost of the other car. 

The equation that would best illustrate the given above is,
                        x + 38295 = 71482

What number is 100 times as much as 340?

Answers

it would be 34000 cos u basically put 2 zeroes on the end
The answer is 100 x 340= 34,000

One book cost £1.99 how much do four books cost

Answers

if 1 book costs 1.99.....then 4 books cost : (4 * 1.99) = $ 7.96 <==
[tex]1.99(4) = 4(2-0.01)[/tex]
[tex]4(2-0.01) = 8 - 0.04 = 7.96[/tex]

Shantel is studying for her history test. After 20 minutes she is 1/4 of the way done. How much longer will she study?

Answers

Shantel will need to study for 60 minutes more to complete her history review.

Shantel is studying history and after 20 minutes, she is [tex]\frac{1}{4}[/tex]  of the way done.

To determine how much longer she will study, we need to calculate how much time is needed to complete the remaining [tex]\frac{3}{4}[/tex] of her study time.

[tex]\frac{1}{4}[/tex] of the study done in 20 minutes means she will study for 4 * 20 = 80 minutes in total.

To find out how much longer she needs to study, subtract the time she has already studied (20 minutes) from the total study time (80 minutes): 80 - 20 = 60 minutes.

Therefore, Shantel will need to study for 60 minutes more to complete her history review.

write each rate as a fraction in lowest terms.

4.20 for 36 potatoes

Answers

$4.20 / 36 = $0.11667 per potato
Something appears to be missing here.  Did you mean $4.20 for 36 potatoes?

If so, write the fraction

$4.20
----------------
36 potatoes

and reduce it.  6 divides into both $4.20 and 36 without a remainder:

$0.70 and 6

Thus, the "rate" is 0.70 dollar for 6 potatoes, or 0.35 dollar for three.

If you wish you could rewrite this partially in words:

"Potatoes sell for $0.35/(3 potatoes)," or $0.1166 per potato, or about 12 cents per potato.

What is the median of the following string of values?
55, 18, 58, 49, 8, 77, 62, 26, 7, 91

A.55

B.52

C.49

D.58

Answers

the median is 52, so the answer would be B.

Answer:

Option B is correct

The median of the following string of values is 52

Step-by-step explanation:

Median defined as the median is the middle number.

To find the median of the following string of values:

55, 18, 58, 49, 8, 77, 62, 26, 7, 91

Arrange the string of  values from least to greatest.

7, 8, 18 , 26, 49, 55, 58, 62, 77, 91

As you can see that we have two numbers, there is no middle number.

Simply, take the mean of the two middle numbers.

then;

Median = [tex]\frac{49+55}{2} = \frac{104}{2} = 52[/tex]

Therefore, the median of the following string of values is 52


If u, v, and w are nonzero vectors in r 2 , is w a linear combination of u and v?

Answers

Not necessarily. [tex]\mathbf u[/tex] and [tex]\mathbf v[/tex] may be linearly dependent, so that their span forms a subspace of [tex]\mathbb R^2[/tex] that does not contain every vector in [tex]\mathbb R^2[/tex].

For example, we could have [tex]\mathbf u=(0,1)[/tex] and [tex]\mathbf v=(0,-1)[/tex]. Any vector [tex]\mathbf w[/tex] of the form [tex](r,0)[/tex], where [tex]r\neq0[/tex], is impossible to obtain as a linear combination of these [tex]\mathbf u[/tex] and [tex]\mathbf v[/tex], since

[tex]c_1\mathbf u+c_2\mathbf v=(0,c_1)+(0,-c_2)=(0,c_1-c_2)\neq(r,0)[/tex]

unless [tex]r=0[/tex] and [tex]c_1=c_2[/tex].

A reel of electrical cable holds 80 metres when full.

3/16 of the reel has been used.

How much cable is there left on the reel?

Answers

I3/I6(8O)= 65m of electrical cable left

How do I solve this 6x + 3 = 5x + 10

Answers

Subtract 5x from both sides
x+3=10

Subtract 3 from both sides
x=7

Final answer: x=7
Move all terms containing xx to the left side of the equation.Since 5x5x contains the variable to solve for, move it to the left side of the equation by subtracting 5x5x from both sides.6x−3−5x=−56x-3-5x=-5Subtract 5x5x from 6x6x to get xx.x−3=−5x-3=-5Move all terms not containing xx to the right side of the equation.Since −3-3 does not contain the variable to solve for, move it to the right side of the equation by adding 33 to both sides.x=3−5x=3-5Subtract 55 from 33 to get −2-2.x=−2

8 to the power of x+3=16 to the power of x-1

Answers

8^x+3=16^x+1, 2^3(x+3)=2^4(x+1), 2^3x+9=2^4x+4, same bases. 3x+9=4x+4, 3x-4x=4-9, -x=-5, x=5
[tex]\bf 8^{x+3}=16^{x-1}\qquad \begin{cases} 8=2\cdot 2\cdot 2\\ \qquad 2^3\\ 16=2\cdot 2\cdot 2\cdot 2\\ \qquad 2^4 \end{cases}\implies (2^3)^{x+3}=(2^4)^{x-1} \\\\\\ 2^{3(x+3)}=2^{4(x-1)}\impliedby \begin{array}{llll} \textit{since the bases are the same}\\ \textit{the exponents must also be the same} \end{array} \\\\\\ 3(x+3)=4(x-1)\implies 3x+9=4x-4\implies 9+4=4x-3x \\\\\\ \boxed{13=x}[/tex]

720,808 in expanded form with exponents

Answers

Wow!  Lot of figuring for us to do here!
You'll need to determine how many times 2 is a factor of 720808, how many times 3 is a factor, and so on.

Suppose that 2 were a factor 6 times (just an example).

You'd write this particular factor of 720808 as 2^6.

The number 720,808 in expanded form with exponents is written as:

[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]

To express 720,808 in expanded form with exponents, We break it down by place value and then represent each term using powers of 10. The breakdown is as follows:

700,000  : [tex]7 \times 10^5 \\[/tex]

20,000  : [tex]2 \times 10^4 \\[/tex]

800  : [tex]8 \times 10^2 \\[/tex]

8  : [tex]8 \times 10^0[/tex]

When we combine these, we get the expanded form:

[tex]\[ 7 \times 10^5 + 2 \times 10^4 + 8 \times 10^2 + 8 \times 10^0 \][/tex]

This representation helps in understanding the value of each digit based on its position in the number.

what are two practical products you could add to find 990 x 37 ; 297 693 2970 6930 69300 6330 29700

Answers

First evaluate 990 x 37.
990 x 37 = 36630

Test the given numbers to find the pair that adds up to 36630.
Test 297.
The missing pair will be
36630 - 297 = 36333
Will not work

Test 693.
36630 - 693 = 35937
Will not work

Test 2970.
36630 - 2970 = 33660
Will not work

Test 6930
36630 - 6930 = 29700
This works, because 6930 + 29700 = 36630.

Answer: 6930 and 29700.


2. During a 5-day training at your department, the number of visitors tripled each day. If the training started on a Monday with 345 attendees, what was the attendance on Thursday?

Answers

Monday/345
345×3=
Tuesday/1,035
1,035×3=
Wendsday/3,105
3,105×3
Thursday=9,315

The number of attendance on Thursday would be 9,315.

Given that,

During a 5-day training at your department, the number of visitors tripled each day.

Here, the training started on a Monday with 345 attendees.

Hence, the number of visitors on Tuesday is 3 times with 345 attendees.

That is, On Tuesday;

Number of attendees = 345 x 3

                                     = 1,035

On Wednesday;

Number of attendees = 1,035 x 3

                                     = 3,105

On Thursday;

Number of attendees = 3,105 x 3

                                     = 9,315

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Derivative of sin (x^(1/3)) + (sinx)^(1/3)

Answers

1/3 x^(-2/3) cos (x^1/3) + 1/3(sinx)^-2/3 .cos x



[tex]\bf sin\left( x^{\frac{1}{3}} \right)+[sin(x)]^{\frac{1}{3}}\\\\ -------------------------------\\\\ cos\left( x^{\frac{1}{3}} \right)\cdot \cfrac{1}{3}x^{-\frac{2}{3}}+\cfrac{1}{3}[sin(x)]^{-\frac{2}{3}}\cdot cos(x)\impliedby \textit{chain-ruling both terms} \\\\\\ \cfrac{cos(\sqrt[3]{x})}{3\sqrt[3]{x^2}}+\cfrac{cos(x)}{3\sqrt[3]{sin^2(x)}}[/tex]

water weighs about 8.34 pounds per gallon. About how many ounces per gallon is the weight of the water?

Answers

Use a conversion factor and turn pounds to ounces. 1 pound = 16 ounces so
[tex] \frac{8.34pounds}{gallon} \times \frac{16 ounces}{ 1 pound} =133.44 \frac{ounces}{gallon} [/tex]

What is the next letter? YBWDUF

Answers

the pattern looks like it is odd terms go back 2 from the second last letter in the alpha bet.
even terms go forward 2 from the second letter in the alpha bet.
so the answer would be S,H,Q,J
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