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tatyana61 [14]
3 years ago
10

Which expression is a cube root of -1+i sqrt3

Mathematics
1 answer:
Marina CMI [18]3 years ago
4 0
If z=-1+i \sqrt{3}, then 

Rez=-1
Imz= \sqrt{3} and 
|z|= \sqrt{Re^2z+Im^2z}= \sqrt{1^2+ (\sqrt{3})^2 }= \sqrt{4}=2.
Then cos\theta = \frac{Rez}{|z|} = \frac{-1}{2}sin\theta = \frac{Imz}{|z|}= \frac{ \sqrt{3} }{2}.

Since \theta \in (-\pi,\pi], you can conclude that \theta= \frac{2\pi}{3}.
The triginometric form of z is z=|z|(cos\theta+isin\theta)=2(cos \frac{2\pi}{3} +isin \frac{2\pi}{3} ).
Use formula \sqrt[3]{z} =\{ \sqrt[3]{|z|} (cos \frac{\theta+2\pi 
k}{n}+isin \frac{\theta+2\pi k}{n} ), k=0,1,2\} to find cube root. 
For k=0, z_1= \sqrt[3]{2} (cos \frac{ 
\frac{2\pi}{3} }{3} +isin\frac{ \frac{2\pi}{3} }{3})= \sqrt[3]{2} (cos 
40^0+isin 40^0),
For k=1, z_1= \sqrt[3]{2} (cos
 \frac{ \frac{2\pi}{3}+2\pi }{3} +isin\frac{ \frac{2\pi}{3}+2\pi }{3})= 
\sqrt[3]{2} (cos 160^0+isin 160^0),
For k=2, z_1= \sqrt[3]{2} (cos \frac{ \frac{2\pi}{3}+4\pi }{3} +isin\frac{ 
\frac{2\pi}{3}+4\pi }{3})= \sqrt[3]{2} (cos 280^0+isin 280^0).
Answer: The correct choice is C.
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A triangle has sides with lengths of 9 yards, 12 yards, and 15 yards. Is it a right triangle?
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Answer:

Yes

Step-by-step explanation:

Plug your values into the Pythagorean Theorem to check if they make sense

3 0
3 years ago
Which expression is equivalent to (-3)x+ (-3) (-12)?
julia-pushkina [17]

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A

Step-by-step explanation:

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6 0
2 years ago
Humberly has $24 and they want to buy at least 8 containers of yogurt. Yogurt is packaged in both large and small containers. La
yanalaym [24]

Answer:

Hence he will be 4 large containers and 4 small containers

Step-by-step explanation:

Given data

Let the number of small containers be x

and the number of large containers be y

x+y= 8---------1

also

2x+4y= 24-----2

the system of equation to solve the problem is

x+y= 8

2x+4y= 24

from 1

x=8-y

put this in 2

2(8-y)+4y= 24

16-2y+4y= 24

2y= 24-16

2y= 8

y= 8/2

y= 4

put y= 4 in 1

x+4=8

x= 8-4

x= 4

Hence he will be 4 large containers and 4 small containers

6 0
3 years ago
Write the cubic polynomial function f(x)in expanded form with zeros −6,−5,and −1, given that f(0)=60
RSB [31]

Answer:

f(x)=2x^{3}+24x^{2}+82x+60

Step-by-step explanation:

we know that

The roots of the polynomial are the values of x when the value of the polynomial f(x) is equal to zero

The roots of the polynomial function are

x=-6 -----> (x+6)=0

x=-5 -----> (x+5)=0

x=-1 -----> (x+1)=0

The equation of the cubic polynomial is

f(x)=a(x+6)(x+5)(x+1)

where

a is the leading coefficient

Remember that

f(0)=60

That means ------> For x=0 the value of f(x) is equal to 60

substitute the value of x and the value of y in the function and solve for a

60=a(0+6)(0+5)(0+1)

60=a(6)(5)(1)

60=30a

a=2

so

f(x)=2(x+6)(x+5)(x+1)

Applying the distributive property

Convert to expanded form                  

f(x)=2(x+6)(x+5)(x+1)\\\\f(x)=2(x+6)(x^{2}+x+5x+5)\\\\f(x)=2(x+6)(x^{2}+6x+5)\\\\f(x)=2(x^{3}+6x^{2}+5x+6x^{2}+36x+30)\\\\f(x)=2x^{3}+24x^{2}+82x+60

3 0
3 years ago
This question is confusing. Please help me!
velikii [3]

Answer:

x = 18

Step-by-step explanation:

Given:

  • a = 2x +3
  • b = 4x - 7

Finding x for which 3b = 5a:

  • 3b = 5a
  • 3(4x - 7) = 5(2x + 3)
  • 12x - 21 = 10x + 15
  • 12x - 10x = 15 + 21
  • 2x = 36
  • x = 36/2
  • x = 18
7 0
3 years ago
Read 2 more answers
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