Answer and step-by-step explanation:
The polar form of a complex number 
is the number 
where 
is called the modulus and 
is called the argument. You can switch back and forth between the two forms by either remembering the definitions or by graphing the number on Gauss plane. The advantage of using polar form is that when you multiply, divide or raise complex numbers in polar form you just multiply modules and add arguments.
(a) let's first calculate moduli and arguments
now we can write the two numbers as
(b) As noted above, the argument of the product is the sum of the arguments of the two numbers:
(c) Similarly, when raising a complex number to any power, you raise the modulus to that power, and then multiply the argument for that value.
![(z_1)^1^2=[4e^{-i\frac \pi6}]^1^2=4^1^2\cdot (e^{-i\frac \pi6})^1^2=2^2^4\cdot e^{-i(12)\frac\pi6}\\=2^2^4 e^{-i\cdot2\pi}=2^2^4](https://tex.z-dn.net/?f=%28z_1%29%5E1%5E2%3D%5B4e%5E%7B-i%5Cfrac%20%5Cpi6%7D%5D%5E1%5E2%3D4%5E1%5E2%5Ccdot%20%28e%5E%7B-i%5Cfrac%20%5Cpi6%7D%29%5E1%5E2%3D2%5E2%5E4%5Ccdot%20e%5E%7B-i%2812%29%5Cfrac%5Cpi6%7D%5C%5C%3D2%5E2%5E4%20e%5E%7B-i%5Ccdot2%5Cpi%7D%3D2%5E2%5E4)
Now, in the last step I've used the fact that 
, or in other words, the complex exponential is periodic with 
as a period, same as sine and cosine. You can further compute that power of two with the help of a calculator, it is around 16 million, or leave it as is.
Hi there! The answer is 10.
We can use PEMDAS (Parenthesis, exponents, multiply, divide, add, subtract) to find our answer.
Since we cannot work inside the parenthesis in this question and since we don't have exponents, we can move on to multiplying.
Our next step would be subtracting. To make the process of subtracting some easier, I'll first change the fraction into a decimal.
Finally subtract.
The answer is 10.
~ Hope this helps you!
Answer:
Red = 30 drops
Blue = 20 drops
Step-by-step explanation:
The ratio red to blue is 3 : 2
Sum of ratios = 3 + 2 = 5
Required purple paint = 50 gallons
So,
Now, Red drops = 3×10 = 30
Blue drops = 2×10 = 20
Answer:
Thus, the two root of the given quadratic equation 
is 2 and -3 .
Step-by-step explanation:
Consider, the given Quadratic equation,
This can be written as , 
We have to solve using quadratic formula,
For a given quadratic equation 
we can find roots using,
...........(1)
Where, 
is the discriminant.
Here, a = 1 , b = 1 , c = -6
Substitute in (1) , we get,
and 
and 
and 
Thus, the two root of the given quadratic equation is 2 and -3 .
