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Tutors Answer Your Questions about Complex Numbers (FREE)
Question 240798: my question is what is the square root of -12 divided by the square root of -3? im in pre calc and it has been a long time since i had to do this so if you could also provide a good explanation of complex numbers that would be great my teacher gave me notes but she didnt give any explanation because she wasnt there so they dont make any sense to me
Answer by edjones(3299)   (Show Source):
Question 240605: (4-3i)-(7+8i)
Answer by stanbon(26297)  (Show Source):
Question 240571: 2-i/3-4i
Answer by jim_thompson5910(13794) (Show Source):
You can put this solution on YOUR website! Start with the given expression.
 Multiply the fraction by  .
 Combine the fractions.
 FOIL the numerator.
 FOIL the denominator.
 Multiply.
 Replace  with -1.
 Multiply.
 Combine like terms.
 Break up the fraction.
 Reduce.
So  .
So the expression is now in standard form  where  and 
Question 240496: How can you know how many number there are in a given sequence?
For example from 1 to 100, there are 100 numbers
From 100 to 1000 there are 901 numbers(I know because my teacher tells me)
How can you know right away how many numbers there are?
Please explain a little bit carefully! Thanks for your helps!
Answer by jsmallt9(594) (Show Source):
You can put this solution on YOUR website!In general, the formula is: last - first + 1.
To understand this, let's look at the example you gave: How many numbers from 100 to 1000? - First, how many numbers are there from 1 to 1000? Clearly (I hope) there are 1000 numbers.
- Of the numbers from 1 to 1000, which ones are not included in the numbers from 100 to 1000? Again I hope it is clear that 1 through 99 are not included.
- The number of numbers from 100 to 1000 would be the number of numbers of 1 to 1000 minus the number of numbers from 1 to 99: 1000 - 99 = 901.
Let's look at a general case. Let L = the last (or highest number) and F = the first (or lowest number in the sequence. That makes (F-1) the highest number that is not in the sequence.
Therefore the number of numbers in the sequence will be: the number of numbers from 1 to L (which is L) minus the number of numbers from 1 to (F-1) (which is (F-1):
L - (F-1) = L - F + 1
which is the formula I gave you at the beginning.
Here's an example: How many numbers are there from 536 to 2349? Answer: 2349 - 536 + 1 = 1814
I hope this helps.
Question 240201: hi again =]
i have a similiar question to the previous one, using the property of exponents can you write the complex number in standard form?
2+i^2
and
5-i^4
thank you!
Answer by solver91311(5072)  (Show Source):
Question 240195: using the property of exponents can you write the complex number in standard form please =]
3+i^3
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!using the property of exponents can you write the complex number in standard form please =]
3+i^3
--------------
i^3 = (i)^2*i = -i
-------------------
3+i^3 = 3-i
===================
Cheers,
Stan H.
Question 239998: 5x^2+500=0
Answer by Alan3354(6097)  (Show Source):
Question 239948: I have a couple of questions that I am stuck with...
(1/4x+3)squared = 48
options are
A.{-12+16 suared root3}, -12-16squareroot 3}
B.{12+16square root 3, 12-16square root 3}
C.{16squareroot3 - 12, 16square root 3+12}
D.{-3+4square root3 over 4, -3-4square root 3 over 4}
Square root 3x+10 = 5 - 2x
options are
A. 3/4
B. 5/4, 9
C. 5
D.3/4, 5
Last one is
square root 2x+5 - square root x-2 = 3
Please help. Thank you
Answer by checkley77(7072) (Show Source):
You can put this solution on YOUR website!(X/4+3)^2=48
X^2/16+3X/2+9=48
X^2/16+3X/2=48-9
(X^2+3X*8)/16=39
X^2+24X=624
X^2+24X-624=0
X=(-24+-SQRT[24^2-4*1*-624])/2*1
X=(-24+-SQRT[576+2496])/2
X=(-24+-SQRT3072)/2
X=(-24+-55.426)/2
X=(-12+-16SQRT3) ANSWER A).
-------------------------------------------
Square root 3x+10=5-2x
SQRT(3X+10)=5-2X SQUARE BOTH SIDES
3X+10=(5-2X)^2
3X+10=25-20X+4X^2
4X^2-20X-3X+25-10=0
4X^2-23X+15=0
(4X+3)(X-5)=0
4X+3=0
4X=-3
X=-3/4 ANS.
X-5=0
X=5 ANS.
SOLUTION (3/4,5) ANSWER D)
-----------------------------------
square root 2x+5 - square root x-2 = 3
SQRT(2X+5)-SQRT(X-2)=3 SQUARE BOTH SIDES.
2X+5+X-2=9
3X=9-5+2
3X=6
X=6/3
X=2 ANS.
PROOF:
SQRT(2*2+5)-SQRT(2-2)=3
SQRT(4+5)-0=3
SQRT9=3
3=3
Question 239677: I have an algebra II quiz on complex numbers and im not sure if i understand any of it...
this is a problem I attempted to work on but im not sure if I am doing the right things in order to solve it out.
heres the equation/problem:
(2+3i) + (7+i)
14+3i2<---( i squared)
14-3
Answer: 11
Answer by ankor@dixie-net.com(6693)  (Show Source):
You can put this solution on YOUR website!(2+3i) + (7+i)
:
Not sure what you are doing here
It's plus, (not multiply) so just remove the brackets and add like terms:
2 + 3i + 7 + i
group like terms
2 + 7 + 3i + i
9 + 4i
Question 239709: 3s squared +2 = -62
I subtracted 2 so that 3s squared = -60 and divided -60 by 3, giving me -20. I then had s squared = -20 and, since you can't have negative square roots, i put S= + or - 2 root 5, but im not sure if that is right.
Answer by edjones(3299) (Show Source):
You can put this solution on YOUR website!3s^2+2=-62
3s^2=-64
s^2=-64/3
s=+-8sqrt(-1/3)
=8sqrt(1/3)i [i=sqrt(-1)]
sqrt(1/3)=1/sqrt(3)=1/sqrt(3) * sqrt(3)/sqrt(3) = sqrt(3)/3 no sqrt in denominator.
s=8sqrt(3)i/3, s=-8sqrt(3)i/3
.
Ed
Question 239713: (3+ root negative four) (4+ root negative 1)
i made the root negative four in the first parenthesis i root 2 and the negative root i in the second parenthesis 4i. im not sure what to do next thanks
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!(3+ root negative four) (4+ root negative 1)
i made the root negative four in the first parenthesis i root 2 and the negative root i in the second parenthesis 4i. im not sure what to do next thanks
--------------------------------------------------------
Good start:
(3 + 2i)(4 + i)
---
Use FOIL to get:
F: 3*4 = 12
O: 3i
I: 8i
L: -2
-------------------
Combine to get: 10+11i
=============================
Cheers,
Stan H.
Question 239615: How do you solve x^2+4? Is it 2i or +/- 2i?
Thanks.
Answer by solver91311(5072) (Show Source):
Question 238836: Hello,Well im stuck on this Quadratic formula.I really need your help, because I tried to do it my self but im not doing it right I guess because the answer im getting is not the answers provided. Can you help me please?
Problem:x^2-8x+30=0
My work:a=1, b=-8, c=30
x=[8+-(-8^(2)-4(1)(30))^(1/2)]/2(1)
=[8+-(64-120)^(1/2)]/2
=[8+-(-56)^(1/2)]/2
=[8+-2i(14)^(1/2)]/2
=4+-i(14)^(1/2)
My options:
A.) -4+4i,-4-4i
B.)4+4i,4-4i
C.)4+8i,4-8i
D.)-4+8i,-4-8i
Answer by ankor@dixie-net.com(6693) (Show Source):
You can put this solution on YOUR website!Problem:x^2 -8x + 30 = 0
My work:a=1, b=-8, c=30
:
:
:
:
:
:
 ; I got the same thing you got
:
Hey!
We can prove it, find the equation using our solution:
:
:
Square both sides
(x-4)^2 = i^2(14)
Which is
x^2 - 8x + 16 = -1(14)
x^2 - 8x + 16 = -14
x^2 - 8x + 16 + 14 = 0
x^2 - 8x + 30 = 0; does that look familiar?
Question 238815: Can you please help me sove this problem. I use the quadratic formula, but I get stuck so maybe that's not the method to use on this problem. Can you help me please?
Answer by Alan3354(6097) (Show Source):
Question 238741: Use de Moivre's Theorem to find the square root of -5+12i.
r=13 √ 25+144 = √ 169
arctan= (12 ÷ 5) = 1.1760...
13(cos 1.7600... +2πn + isin 1.1760...)
13^(1/2)(((cos 1.1760... + 2πn)/2)+((isin1.76)/2)
This is what I have done so far, please tell me if it is correct, and I am unsure as to what steps to make next.
Answer by Alan3354(6097) (Show Source):
You can put this solution on YOUR website!Use de Moivre's Theorem to find the square root of -5+12i.
r=13 √ 25+144 = √ 169
------------------------
angle A = arccos(-5/13)
z = 13(cos(A) + isin(A)) A =~ 112.62º
sqrt(z) = sqrt(13)*(cos(A/2) + isin(A/2))
sqrt(z) = sqrt(13)*(0.5547 + i*0.8321)
= 2 + 3i
----------
arctan= (12 ÷ 5) = 1.1760...
13(cos 1.7600... +2πn + isin 1.1760...)
13^(1/2)(((cos 1.1760... + 2πn)/2)+((isin1.76)/2)
Question 238734: how do you find the square root of 25/3 using complex numbers?
Found 2 solutions by stanbon, checkley77:
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!how do you find the square root of 25/3 using complex numbers?
-----------------
25/3 = 25/3 + 0i
---------------------
Trigonometric Form:
25/3+0i = (25/3)(cos(0)+isin(0))
-------------------------------
To take the square root:
(25/3)^(1/2) = (25/3)^1/2(cos[(0+2npi)/2] + isin([0+2npi)/2] for n = 0,1
---------------------
For n=0 you get:
(25/3)^(1/2)(cos(0) + isin(0) = sqrt(25/3) + 0i
------------------------------
For n=1 you get:
(25/3)^(1/2)(cos(pi)+isin(pi)) = -sqrt(25/3)+0i
=====================================================
That is the complex number answer but I think you really
may have wanted the irrational number answer which is
as follows:
---
sqrt(25/3) = 5sqrt(1/3) = 5sqrt(3/9) = 5sqrt(3)/3
= (5/3)sqrt(3)
=====================================================
Cheers,
Stan H.
Answer by checkley77(7072) (Show Source):
Question 238652: Evaluate  by using De Moivre's Theorem. Express the result in rectangular form.
Thank you!
Answer by jim_thompson5910(13794) (Show Source):
You can put this solution on YOUR website!First you have to convert  into polar form +i\sin(\theta)\right)) .
To do that, use the formulas  and )
In this case, we're given  which means that  and  . So  and =\tan^{-1}\left(1\right)=\frac{\pi}{4})
So this means that the rectangular expression  is equivalent to the polar form +i\sin(\frac{\pi}{4})\right)) .
From here, we can now use De Moivre's Theorem. De Moivre's Theorem states that if , then +i\sin(n\theta)\right))
Since , using De Moivre's Theorem gets us
^{12}\left(\cos(12*\frac{\pi}{4})+i\sin(12*\frac{\pi}{4})\right))
+i\sin(3\pi)\right)) ... Evaluate  to the 12th power to get 
+i\sin(3\pi)\right)) ... Evaluate 2 to the 6th power to get 64
) ... Evaluate the trig functions.
) ... Simplify.
 ... Multiply
Because we let , this means that 
Question 238358: Please help me solve this equation (-5-2i)(3+7i)
Answer by stanbon(26297) (Show Source):
Question 238277: 
i dont understand the whole i thing
Found 2 solutions by College Student, checkley77:
Answer by College Student(217)  (Show Source):
You can put this solution on YOUR website!Ok. If we try to solve for x, this is what we would get:
 ...same as...  (an imaginary answer)
.
Now, we know that if we multiply two negative numbers, the result will always be a positive number. But in this problem, we see that a number raised to the second power (same as having a number multiplied by itself) should equal -3. This directly contradicts the rule. So, Rafael Bombelli (back in 1572) invented the imaginary number we know as "i" to be able to work out these problems.
.
"i" = 
.
Suppose the answer would have been  ... since we know the square root of 9 is 3, we could then say  ...same as... 
But in your problem, leave it as .
.
I hope this explanation helped. :)
Answer by checkley77(7072) (Show Source):
Question 238211: combine terms
-3c-4b+2a-b-c-7a
how do i solve this?
Answer by nyc_function(260)  (Show Source):
Question 238026: solve for x and y
(5+2i)+(x+yi)=4-3i
Answer by MissionPossible(18) (Show Source):
Question 237884: Prove how 3 is irrational
Answer by vleith(1977) (Show Source):
Question 236724: i really dont know but its about remain restriction and could you please help me?
the problem is:
Y=5X/3X+9?
can you please help me?
Found 2 solutions by solver91311, College Student:
Answer by solver91311(5072) (Show Source):
Answer by College Student(217) (Show Source):
You can put this solution on YOUR website!If I understand correctly, you're trying to find the value of Y, right?
Since you have an x to the power of 1 in the numerator and in the denominator, we can cancel the x from the equation. So:
This is the same as to say:
So now we look for a common denominator, which would be 3.
Now you can add the fractions as they have same denominator.
Question 237233: Atmospheric pressure P at an elevation of a feet above sea level is given by
P=P0e-0.00004a
Where P0 is the pressure at sea level,which is approximately 29.9 in inches of mercury(Hg).
Explain how a barometer, or some other device for measuring atmospheric pressure , can be used to find the height of a skyscraper.
Answer by Theo(675)  (Show Source):
You can put this solution on YOUR website!Equation would be:
P[a] =P[0] * e^(-0.00004*a)
* means multiply
^ means raise to the power of
a appears to be altitude
p[0] = 29.9 inches of mercury
p[a] would be the inches of mercury at an altitude of "a" feet.
since p[0] = 29.9 inches of mercury, we can replace p[0] in the equation with 29.9
assuming a = 0, then this equation would become:
p[0] = 29.9 * e^(-.00004*0)
this would become:
p[0] = 29.9 * e^0 which would become p[0] = 29.9 which would be correct since we would be at sea level.
if you were on a structure that was 1000 feet high, then this equation would become:
p[1000] = 29.9 * e^(-.00004*1000) which would become:
p[1000] = 29.9 * .960789439 which would become:
p[1000] = 28.72760423
If you want to measure the height of a building, all you have to do is go to the top of the building and measure the atmospheric pressure.
Once you have that, you can plug the value into the equation and solve for the height.
As an example:
Assume you go to the top of the building and you measure the atmospheric pressure to get:
atmospheric pressure at top of building = 27 inches of mercury.
your equation is:
p[a] = 29.9 * e^(-.00004*a)
Since you already know p[a], then you plug it into the equation to get:
27 = 29.9 * e^(-.00004*a)
divide both sides of this equation by 29.9 to get:
27/29.9 = e^(-.00004*a) which becomes:
.903010033 = e^(-.00004*a)
take the natural log of both sides of this equation to get:
ln(.903010033) = ln(e^(-.00004*a))
using the laws of logarithms, this equation becomes:
ln(.903010033) = ln(e) * (-.00004*a)
since ln(e) = 1, this equation becomes:
ln(.903010033) = -.00004*a
divide both sides of this equation by (-.00004) to get:
ln(.903010033)/(-.00004) = a
solve for a to get:
a = 2550.54036 feet.
the building would be approximately 2550 feet in height.
that's a very tall building, but that's only because I chose 27 inches of mercury at random without really knowing how high that would be.
in reality your measurement of atmospheric pressure would probably be around 28 or higher.
Question 237162: 6+2√3
-----------
8-7√3
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!6+2√3
-----------
8-7√3
0000000000000000000
Multiply numerator and denominator by 8+7Sqrt(3) to get:
0000000000000000000
[(6+2sqrt(3))(8+7sqrt(3))]/[64-49*3]
-----------------------------
= [48 + 58sqrt(3) + 14*3]/[-83]
---
= [90+58sqrt(3)]/(-83)
============================
Cheers,
Stan H.
Question 236939: write the complex number 13 + srq root -9 in standard form.
Answer by edjones(3299) (Show Source):
Question 236941: How would you slove this word problem?
A plane traveled 1080 miles to San Francisco and back. The trip there was with the wind. It took 10 hours. The trip back was into wind. The trip back took 15 hours. What is the speed of the plane in still air? What is the speed of the wind?
Answer by scott8148(3385)  (Show Source):
You can put this solution on YOUR website!rate = distance / time
p + w = 1080 / 10 = 108
p - w = 1080 / 15 = 72
adding equations ___ 2p = 180 ___ p = 90
substituting ___ 90 + w = 108 ___ w = 18
Question 236214: Hello,
I'm to find the third roots of 343i written in polar form with arguments in radians. I don't know how to approach this one. Is it actually "0 + 343i"? If so, I find the absolute value to be 343. Then tan (b/a) = tan (343/0) which is undefined. How do I find the angles?
Please help me as I'm lost...
Answer by Alan3354(6097) (Show Source):
You can put this solution on YOUR website!I'm to find the third roots of 343i written in polar form with arguments in radians. I don't know how to approach this one. Is it actually "0 + 343i"? If so, I find the absolute value to be 343. Then tan (b/a) = tan (343/0) which is undefined. How do I find the angles?
--------------
It is 0 + 343i, and r = 343.
The angle is 90, but you don't need the tangent
z = 0 + 343i = 343cis90
--------
Cube roots are:
7cis30 = 7cis(pi/6) = 7cos(30) + 7isin(30) = 7sqrt(3)/3 + 3i/2
---------------------
Do these 2 the same way.
7cis150 = 7cis(5pi/6)
7cis270 = 7cis(2pi/3)
Question 235915: Show a complete solution.
Big plane. A six-passenger plane cruises at 180 mph in
calm air. If the plane flies 7 miles with the wind in the
same amount of time as it flies 5 miles against the wind,
then what is the wind speed?
Answer by checkley77(7072) (Show Source):
You can put this solution on YOUR website!D=RT OR T=D/R
T=7/(180+W)
T=5/(180-W) SEEING AS THE TIMES ARE EQUAL WE SET THESE 2 EQUATION EQUAL
7/(180+W)=5/(180-W)
7(180-W)=5(180+W)
1,260-7W=900+5W
-7W-5W=900-1,260
-12W=-360
W=-360/-12
W=30 MPH.IS THE WIND SPEED.
PROOF:
7/(180+30)=5/(180-30)
7/210=5/150
1/30=1/30
Question 235955: 3 roots of 28x squared y cubed t to the 11
do you take out 2X'S, 3y's , and 3 t's first
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!3 roots of 28x squared y cubed t to the 11
--------------
Not sure what you mean by "3 roots".
Do you mean 3 time the sqrt(28x^2y^3t^11)
---
or do you mean the cube root of 28x^2y^3t^11 ??
Cheers,
Stan H.
Question 235440: I am having problem with this problem complex simplifying fraction.
2x-4/xy^2/3x-6y/x^3y
Answer by ankor@dixie-net.com(6693) (Show Source):
Question 235867: how can you show that √3+i/2 is a cube root of i
Answer by josmiceli(3013) (Show Source):
You can put this solution on YOUR website!Use the Argand plane where multiplication
is done by multiplying amplitudes and adding
angles of the rotating vectors
 =  angle 30 degrees
angle 30 degrees =  angle 30 degrees
 = angle 90 degrees
angle 90 degrees =  angle  degrees
angle 90 degrees = angle 90 degrees
It is best to draw this. The advantage is that the exponent could
be any size and you just add angles and raise 1 number to a power
Question 235444: This is a word problem, I am terrible with word problems.
After leaving Moose Jaw, Hanson drove
200 kilometers at x km/hr and then decreased his speed by
20 km/hr and drove 240 additional kilometers. Make a table
like the one in Exercise 87.Write a rational expression for
his total traveling time. Evaluate the expression for x = 100.ms.
Answer by josmiceli(3013) (Show Source):
You can put this solution on YOUR website!Each change requires a different
equation because a change will make
the graph of  a different
straight line
1st part of trip:
2nd part of trip:
This is 2 equations and 2 unknowns, so
it's solvable
My tip for word problems is: A small amount of time
spent really thinking about what's going on in
general is worth doing a hundred problems.
Question 235195: I think that my problem falls under this topic. I am in Algebra 2 and I am stuck on this one problem. Solving Systems of Equations in Three Variables.
Here's the problem:
1.)2x-y+z=1
2.)x+2y-4z=3
3.)4x+3y-7z=-8
So here's what I tried to do:
First, I put equations 1 and 2 together and used the elimination method and I got -5y+9z=-5
so that would be the 4th equation
Then, I put equations 2 and 3 together and used elimination again and got -5y+9z=-20
which would be the 5th equation
I tried to put equations 4 and 5 together since they had like terms but it just didn't seem to work out well at all. Asking for your help and thanking you in advance, bthnydyson
Answer by ankor@dixie-net.com(6693) (Show Source):
You can put this solution on YOUR website!1.)2x - y + z = 1
2.)x + 2y -4z = 3
3.)4x +3y -7z =-8
:
Multiply the 1st equation by 2 and add to the 2nd equation
4x - 2y + 2z = 2
x + 2y - 4z = 3
-------------------addition eliminates y
5x - 2z = 5
;
Multiply the 1st equation by 3 and add to the 3rd equaiont
6x - 3y + 3z = 3
4x + 3y - 7z = -8
--------------------addition eliminates y again
10x - 4z = -5
:
You can see right now that this system has no solution
If you multiply eq 4 by 2 and add to the above, you can see
10x - 4z = 10
10x - 4z = -5
:
No wonder you were having trouble
Question 235247: what is the square root of 32 in simple radical form?
Answer by Alan3354(6097) (Show Source):
Question 235046: can someone please help show me how to sovle the following problem? i^37
Answer by Alan3354(6097) (Show Source):
You can put this solution on YOUR website!i^37
= i
--------
i^2 = -1
i^4 = 1
i^8 = 1
Any multiple of 4 in the exponent --> 1
--------------
37 = 9*4 + 1
Divide the exponent by 4 and use the remainder
i^37 = i^1
= i
Question 235003: 2x - 4x + 3 = 0
-3 + 4x - 2x = 0
in change
Answer by edjones(3299) (Show Source):
Question 234977: prove 2^6cos^4teta(angle)sin^3teta(angle)=-sin7teta(angle)-sin5teta(angle)+3sin3teta(angle)
3sinteta(angle).
Answer by Alan3354(6097) (Show Source):
You can put this solution on YOUR website! prove 2^6cos^4teta(angle)sin^3teta(angle)=-sin7teta(angle)-sin5teta(angle)+3sin3teta(angle)
3sinteta(angle).
------------
That's not clear or decipherable.
2^6cos^4 ???
Question 234733: solve and express the solution in simplest radical form
x^2-12x+25=0
Answer by checkley77(7072) (Show Source):
You can put this solution on YOUR website!x^2-12x+25=0
X=(12+-SQRT[-12^2-4*1*25])/2
X=(12+-SQRT[144-100])/2
X=(12+-SQRT4)/2
X=(12+-6.633)/2
X=(12+6.633)/2
X=18.633/2
X=9.317 ANS.
X=(12-6.633)/2
X=5.367/2
X=2.6835 ANS.
Question 234630: Rachel spent $16.18 for some cans of dog food cost in 79 cents each and some cans of cat food costing 69 cents each. She bought two more cans of cat food than of dog food. How many cans of each did she buy?
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!Rachel spent $16.18 for some cans of dog food cost in 79 cents each and some cans of cat food costing 69 cents each. She bought two more cans of cat food than of dog food. How many cans of each did she buy?
--------------------------
Let the number of cans of dog food = "x".
---
Then the number of cans of cat food = "x+2".
---
Value equation:
value + value = value
79x + 69(x+2) = 1618 cents
---
79x + 69x + 138 = 1618
148x = 1480
x = 10 (# of cans of dog food)
----
x+2 = 12 (# of cans of cat food)
==================================
Cheers,
Stan H.
Question 234513: How do you factor 2b^2+13b-7
so far i have (2b+ )(b- )
Answer by checkley77(7072) (Show Source):
Question 234401: 6√(-12)
is it 12i√3 or 6(2i√3)
Answer by stanbon(26297) (Show Source):
You can put this solution on YOUR website!6√(-12)
is it 12i√3 or 6(2i√3)
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6sqrt(-12)
= 6*sqrt(-4*3)
= 6*(2i)sqrt(3)
= 12i*sqrt(3)
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Cheers,
Stan H.
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