Helppp and explain pls and ty

Answer:

See Below (Boxed Solutions).

Step-by-step explanation:

We are given the two complex numbers:

[tex]\displaystyle z = \sqrt{3} - i\text{ and } w = 6\left(\cos \frac{5\pi}{12} + i\sin \frac{5\pi}{12}\right)[/tex]

First, convert z to polar form. Recall that polar form of a complex number is:

[tex]z=r\left(\cos \theta + i\sin\theta\right)[/tex]

We will first find its modulus r, which is given by:

[tex]\displaystyle r = |z| = \sqrt{a^2+b^2}[/tex]

In this case, a = √3 and b = -1. Thus, the modulus is:

[tex]r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2[/tex]

Next, find the argument θ in [0, 2π). Recall that:

[tex]\displaystyle \tan \theta = \frac{b}{a}[/tex]

Therefore:

[tex]\displaystyle \theta = \arctan\frac{(-1)}{\sqrt{3}}[/tex]

Evaluate:

[tex]\displaystyle \theta = -\frac{\pi}{6}[/tex]

Since z must be in QIV, using reference angles, the argument will be:

[tex]\displaystyle \theta = \frac{11\pi}{6}[/tex]

Therefore, z in polar form is:

[tex]\displaystyle z=2\left(\cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6}\right)[/tex]

Part A)

Recall that when multiplying two complex numbers z and w:

[tex]zw=r_1\cdot r_2 \left(\cos (\theta _1 + \theta _2) + i\sin(\theta_1 + \theta_2)\right)[/tex]

Therefore:

[tex]\displaystyle zw = (2)(6)\left(\cos\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right) + i\sin\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right)\right)[/tex]

Simplify. Hence, our polar form is:

[tex]\displaystyle\boxed{zw = 12\left(\cos\frac{9\pi}{4} + i\sin \frac{9\pi}{4}\right)}[/tex]

To find the complex form, evaluate:

[tex]\displaystyle zw = 12\cos \frac{9\pi}{4} + i\left(12\sin \frac{9\pi}{4}\right) =\boxed{ 6\sqrt{2} + 6i\sqrt{2}}[/tex]

Part B)

Recall that when raising a complex number to an exponent n:

[tex]\displaystyle z^n = r^n\left(\cos (n\cdot \theta) + i\sin (n\cdot \theta)\right)[/tex]

Therefore:

[tex]\displaystyle z^{10} = r^{10} \left(\cos (10\theta) + i\sin (10\theta)\right)[/tex]

Substitute:

[tex]\displaystyle z^{10} = (2)^{10} \left(\cos \left(10\left(\frac{11\pi}{6}\right)\right) + i\sin \left(10\left(\frac{11\pi}{6}\right)\right)\right)[/tex]

Simplify:

Simplify using coterminal angles. Thus, the polar form is:

[tex]\displaystyle \boxed{z^{10} = 1024\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)}[/tex]

And the complex form is:

[tex]\displaystyle z^{10} = 1024\cos \frac{\pi}{3} + i\left(1024\sin \frac{\pi}{3}\right) = \boxed{512+512i\sqrt{3}}[/tex]

Part C)

Recall that:

[tex]\displaystyle \frac{z}{w} = \frac{r_1}{r_2} \left(\cos (\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right)[/tex]

Therefore:

[tex]\displaystyle \frac{z}{w} = \frac{(2)}{(6)}\left(\cos \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right) + i \sin \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right)\right)[/tex]

Simplify. Hence, our polar form is:

[tex]\displaystyle\boxed{ \frac{z}{w} = \frac{1}{3} \left(\cos \frac{17\pi}{12} + i \sin \frac{17\pi}{12}\right)}[/tex]

And the complex form is:

[tex]\displaystyle \begin{aligned} \frac{z}{w} &= \frac{1}{3} \cos\frac{5\pi}{12} + i \left(\frac{1}{3} \sin \frac{5\pi}{12}\right)\right)\\ \\ &=\frac{1}{3}\left(\frac{\sqrt{2}-\sqrt{6}}{4}\right) + i\left(\frac{1}{3}\left(- \frac{\sqrt{6} + \sqrt{2}}{4}\right)\right) \\ \\ &= \boxed{\frac{\sqrt{2} - \sqrt{6}}{12} -\frac{\sqrt{6}+\sqrt{2}}{12}i}\end{aligned}[/tex]

Part D)

Let a be a cube root of z. Then by definition:

[tex]\displaystyle a^3 = z = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]

From the property in Part B, we know that:

[tex]\displaystyle a^3 = r^3\left(\cos (3\theta) + i\sin(3\theta)\right)[/tex]

Therefore:

[tex]\displaystyle r^3\left(\cos (3\theta) + i\sin (3\theta)\right) = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]

If two complex numbers are equal, their modulus and arguments must be equivalent. Thus:

[tex]\displaystyle r^3 = 2\text{ and } 3\theta = \frac{11\pi}{6}[/tex]

The first equation can be easily solved:

[tex]r=\sqrt[3]{2}[/tex]

For the second equation, 3θ must equal 11π/6 and any other rotation. In other words:

[tex]\displaystyle 3\theta = \frac{11\pi}{6} + 2\pi n\text{ where } n\in \mathbb{Z}[/tex]

Solve for the argument:

[tex]\displaystyle \theta = \frac{11\pi}{18} + \frac{2n\pi}{3} \text{ where } n \in \mathbb{Z}[/tex]

There are three distinct solutions within [0, 2π):

[tex]\displaystyle \theta = \frac{11\pi}{18} , \frac{23\pi}{18}\text{ and } \frac{35\pi}{18}[/tex]

Hence, the three roots are:

[tex]\displaystyle a_1 = \sqrt[3]{2} \left(\cos\frac{11\pi}{18}+ \sin \frac{11\pi}{18}\right) \\ \\ \\ a_2 = \sqrt[3]{2} \left(\cos \frac{23\pi}{18} + i\sin\frac{23\pi}{18}\right) \\ \\ \\ a_3 = \sqrt[3]{2} \left(\cos \frac{35\pi}{18} + i\sin \frac{35\pi}{18}\right)[/tex]

Or, approximately:

[tex]\displaystyle\boxed{ a _ 1\approx -0.4309 + 1.1839i,} \\ \\ \boxed{a_2 \approx -0.8099-0.9652i,} \\ \\ \boxed{a_3\approx 1.2408-0.2188i}[/tex]

Answer:

1320

Step-by-step explanation:

Use the formula for sum of series, s(a) = n/2(2a + (n-1)d)

The terms increase by 6, so d is 6

a is the first term, -56

n is the terms you want to find, 33

Plug in the numbers, 33/2 (2(-56)+(32)6)

Simplify into 33(80)/2 and you get 1320

Answer:

günah(3x) + 2

Step-by-step explanation:

Gösterilen sinüzoidal fonksiyonun denklemini yazınız? A) y = cos x + 2 B) y = cos(3x) + 2 C) y = günah x + 2 D) y =

Answer:

y = sin(3x) + 2

Answer:

0.1009

Step-by-step explanation:

To convert percentage into decimal, you need to divide the percentage by 100

10.09/100

= 0.1009

[tex]\\ \rm\Rrightarrow -30=5(x+1)[/tex]

[tex]\\ \rm\Rrightarrow -30=5x+5[/tex]

[tex]\\ \rm\Rrightarrow 5x=-30-5[/tex]

[tex]\\ \rm\Rrightarrow 5x=-35[/tex]

[tex]\\ \rm\Rrightarrow x=\dfrac{-35}{-5}[/tex]

[tex]\\ \rm\Rrightarrow x=7[/tex]

Answer:

x = -7

Step-by-step explanation:

-30 = 5 (x -1 )

5 ( x + 1 ) =-30

5 (x + 1 ) = - 30

     5            5

x + 1 = -6

x + 1 -1 = -6 -1

x = - 7

Pay per shelf = $3.25

No of shelfs per hour = 5

Total hours per day = 8

Total days to find pay of = 7

= 3.25×5×8×7

= 910

Therefore she is paid $910 after 1 week.

Must click thanks and mark brainliest

Answer:

your answer is 30

Step-by-step explanation:

I hope this help

Answer:

Ekta and Preyal

Step-by-step explanation:

Answer:

y = 10

Step-by-step explanation:

To find the y coordinate of the midpoint, take the y coordinates of the endpoints and average

(0+y)/2 = 5

Multiply each  die by 2

0+y = 10

y = 10

Answer:

x=12

Step-by-step explanation:

LM = RS

9x+27 = 135

Subtract 27 from each side

9x+27-27 =135-27

9x=108

Divide each side by 9

9x/9 = 108/9

x = 12

the boxes are equivalent so the one with a single dash is equal to the other with a single dash.

the one with 2 dashes is equal to the other with 2 dashes so on and so forth

SR=LM

LM=9x+27

RS=135

9x+27=135

so I solve it in my own weird way but you can solve it differently. 135-27=108

108/9=12

so your answer is 12

Step-by-step explanation:

A natural number is a positive whole number.

A whole number is a positive number with no fractions or decimals.

A interger is a whole number negative or positive.

A rational number is a number that terminates or continue with repeating digits.

A irrational number is a number that doesn't terminate or continue with repeating digits.

1. Rational Number

2. Natural,Whole,Interger,Rational

3. Whole,Rational,Interger

4. Rational

5.Irrational

6.Rational

7.Natural,Whole,Interger,Rational

8.Interger,Rational

9.Irrational

The surface area of the rectangular pyramid is the sum of the area of its individual surface. The surface area of the pyramid is 384 square inches.

Each surface of the rectangular pyramid has the shape of a triangle.

So that the area of each surface of the pyramid = [tex]\frac{1}{2}[/tex] x base x height.

From the given question;

The area of one of the two lateral faces = [tex]\frac{1}{2}[/tex] x b x h

                                                                 = [tex]\frac{1}{2}[/tex] x 10 x 15

The area of one of the two lateral faces = 75 square inches

Thus,

the area of the first two given lateral faces = 2 x 75

                                                            = 150 square inches

The area of the first two given lateral faces = 150 square inches

Also,

The area of one of the other two lateral faces = [tex]\frac{1}{2}[/tex] x b x h

                                                                             = [tex]\frac{1}{2}[/tex] x 18 x 13

The area of one of the other two lateral faces =  117 square inches

So that;

the area of the first two other lateral faces = 2 x 117

                                                            = 234 square inches

The area of the first two other lateral faces = 234 square inches

Thus,

the surface area of the pyramid = 150 + 234

                                                              = 384 square inches

Therefore, the surface area of the rectangular pyramid is 384 square inches.

For more clarifications: https://brainly.com/question/23564399

Answer:

Step-by-step explanation:

The surface area of the rectangular pyramid is the sum of the area of its individual surface. The surface area of the pyramid is 384 square inches.

Each surface of the rectangular pyramid has the shape of a triangle.

So that the area of each surface of the pyramid =  x base x height.

From the given question;

The area of one of the two lateral faces =  x b x h

                                                                =  x 10 x 15

The area of one of the two lateral faces = 75 square inches

Thus,

the area of the first two given lateral faces = 2 x 75

                                                           = 150 square inches

The area of the first two given lateral faces = 150 square inches

Also,

The area of one of the other two lateral faces =  x b x h

                                                                            =  x 18 x 13

The area of one of the other two lateral faces =  117 square inches

So that;

the area of the first two other lateral faces = 2 x 117

                                                           = 234 square inches

The area of the first two other lateral faces = 234 square inches

The area of the rectangle is B x H = 18 x 10 = 180 square inches

Thus,

the surface area of the pyramid = 150 + 234+180

                                                             = 564 square inches

Therefore, the surface area of the rectangular pyramid is 564 square inches.

Answer:4 in each bad 2 left over

Step-by-step explanation:

Answer:

4 in each bag and 2 left over

Step-by-step explanation:

divide 14 by 3

3 goes into 14, 4 times

14 - 12 = 2

4 in each bag and then 2 left over

Answer:

first odd integer=x

second odd integer=x+2

third odd integer=x+4

x+x+2+x+4=2097

x+x+x+2+4=2097

3x+6=2097

3x=2097-6

3x=2091

3x/3=2091/3

x=697

therefore, x=697

x+2=697+2=699

x+4=697+4=701

Answer:

abc is a triangle so ,

a is ( 9,6 )

b is ( 9,3 )

and c is ( 3,3 )

9514 1404 393

Answer:

  (d)  Right, scalene

Step-by-step explanation:

The little square in the upper left corner tells you that is a right angle. Any triangle with a right angle is a right triangle. This one is scalene, because the sides are all different lengths.

__

Additional comment

An obtuse triangle cannot be equilateral, and vice versa.

An equilateral triangle has all sides the same length, and all angles the same measure: 60°. It is an acute triangle.

It follows from the definition of the derivative and basic properties of arithmetic. Let f(x) and g(x) be functions. Their derivatives, if the following limits exist, are

[tex]\displaystyle f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}h\text{ and }g'(x)\lim_{h\to0}\frac{g(x+h)-g(x)}h[/tex]

The derivative of f(x) + g(x) is then

[tex]\displaystyle \big(f(x)+g(x)\big)' = \lim_{h\to0}\big(f(x)+g(x)\big) \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{\big(f(x+h)+g(x+h)\big)-\big(f(x)+g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{\big(f(x+h)-f(x)\big)+\big(g(x+h)-g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_{h\to0}\frac{f(x+h)-f(x)}h+\lim_{h\to0}\frac{g(x+h)-g(x)}h \\\\ \big(f(x)+g(x)\big)' = f'(x) + g'(x)[/tex]

Answer:

x(ax-1)-y(y+1)

Step-by-step explanation:

you have to group the like terms

ax^2-x-y^2-y

x(ax-1)-y(y+1)

I hope this helps

Answer:

x = 6.2

Step-by-step explanation:

Since this is a right triangle, we can use trig functions

tan theta = opp / adj

tan 32 = x/ 10

10 tan 32 = x

x=6.24869

Rounding to the nearest tenth

x = 6.2

Answer:

x=6.2 (Rounded to the nearest tenth)

Step-by-step explanation:

This problem gives you an angle(32°), and ask for the dimension of the opposite side to that angle(x), along with another dimension the adjacent side(10).

Since you have the opposite and adjacent sides, you can use tangent. opposite (x) over adjacent (10). Tan(32) =[tex]\frac{x}{10}[/tex]. You want (x) so multiply tan(32) by 10. Then round to the nearest tenth.

Remember to put calculator in degree mode!

tan (32) = 0.6248693519 multiply by ten 6.248693519. Round to nearest 10th 6.2.

Hope this helps!

Step-by-step explanation:

-9x+15=3(2-x)

6-6x

2. collect like terms

-9x+15= 6-6x

15-6 = 6x+9x

11= 15x

3. divide both sides by the coefficient of X which is 15

x= 11/15

Answer:

C

Step-by-step explanation:

as we can see on the graph, the lowest y value is the vertex/corner at x=3, y=-6.

all other y values are above (=are larger) that level.

and it goes up without appearant limit, so up to infinity.

Step-by-step explanation:

V=4/3πr^3

V=4/3π(3.85)^3

V=4/3π(57.066625)

V=4/3 (179.280089865)

V=239.04011982

V=239 ft^3

Answer:

a = -24

Step-by-step explanation:

[tex]30=26-\frac{a}{6}\\(6)4=-\frac{a}{6}(6)\\24=-a\\-24=a[/tex]

Answer:

-24 = a

Step-by-step explanation:

30 - 26 - a/6

Subtract 26 from each side

30 -26 = 26 - a/6 - 26

4 = -a/6

Multiply by -6 on each side

-6 *4 = -a/6 *-6

-24 = a

Answer:

sqrt(3) /3

Step-by-step explanation:

1 / sqrt(3)

Multiply the top and bottom by sqrt(3)

1/ sqrt(3) * sqrt(3)/ sqrt(3)

sqrt(3) /  sqrt(3)*sqrt(3)

sqrt(3) /3

Answer:

[tex] = { \sf{ \frac{1}{ \sqrt{3} } }} \\ \\ { \sf{ = \frac{1}{ \sqrt{3} } . \frac{ \sqrt{3} }{ \sqrt{3} } }} \\ \\ = { \sf{ \frac{ \sqrt{3} }{ {( \sqrt{3}) }^{2} } = \frac{ \sqrt{3} }{3} }} [/tex]

Answer:

Step-by-step explanation:

I am assuming 1 and 2 are asking for factors,

I am only gonna solve 4 for the sake of time,

9x^5-x^3+2x^2-x

x(9x^4-x^2+2x-1)

x(9x^4-(x^2-2x+1)

x(9x^4-(x-1)^2)

x(3x^2+x-1)(3x^2-x+1)

x^4+2x^2-24

x^4+6x^2-4x^2-24

x^2(x^2+6)-4(x^2+6)

(x^2-4)(x^2+6)

(x+2)(x-2)(x^2+6)

a^4+a^2+1

a^4+1+a^2

(a^2+1)^2-2a^2+a^2

(a^2+1)^2-a^2

(a^2+1-a)(a^2+1+a)

(a^2+a+1)(a^2-a+1)

a^3+b^3+c^3-3abc

=(a+b)^3+c^3−3ab(a+b)−3abc

=(a+b+c)^3−(3c(a+b)^2+3(a+b)c^2)−3ab(a+b+c)

=(a+b+c)^3−3c(a+b)(a+b+c)−3ab(a+b+c)

=(a+b+c)^3−(a+b+c)(3ab+3bc+3ac)

=(a+b+c)(a^2+b^2+c^2+2ab+2bc+2ac)−(a+b+c)(3ab+3bc+3ac)

=(a+b+c)(a^2+b^2+c^2−ab−bc−ac)

Answer:

228573

Step-by-step explanation:

a = 219 (first term)

an = 2018 (last term)

Sn->Sum of n terms

Sn=n/2(a + an)         [Where n is no. of terms] -> eq 1

To find number of terms,

an = a + (n-1)d     [d->Common Difference] -> eq 2

d= 226-219 = 7

=> d=7

Substituting in eq 2,

2018 = 219 + (n-1)(7)

1799 = (n-1)(7)

1799 = 7n-7

1799 = 7(n-1)

1799/7 = n-1

257 = n-1

n=258

Substituting values in eq 1,

Sn = 258/2(219+2018)

    = 129(2237)

    = 228573

[tex]\\ \sf\longmapsto 2^3+6^5[/tex]

[tex]\\ \sf\longmapsto 2^3+(2\times 3)^5[/tex]

[tex]\\ \sf\longmapsto 8+2^5\times 3^5[/tex]

[tex]\\ \sf\longmapsto 8+32\times 243[/tex]

[tex]\\ \sf\longmapsto 40+7776[/tex]

[tex]\\ \sf\longmapsto 7784[/tex]

Answer:

2*2*2= 8

6*6*6*6*6= 7,776

7,776+8=

7,784