[100 points]
Using mathematically precise language, explain in detail how you would multiply the complex number z1=r1 (cos theta1+isin theta 1) with the complex number z2=r2 (cos theta 2+isin theta 2)
Answer:
Multiply the moduli and add the arguments.Step-by-step explanation:First write out the two terms being multiplied: z1z2= [r1(cos(theta 1) + isin(theta 1))][r2(cos(theta 2)+ isin(theta 2))] z1z2= r1r2 (cos(theta 1) + isin(theta 1))(cos(theta 2)+ isin(theta 2)) z1z2= r1r2(cos(theta 1)cos(theta 2))+(icos(theta 1)sin(theta 2))+(icos(theta 2)sin(theta 1))+(i^2 sin(theta 1)sin(theta 2)) Next group and rewrite the like term with the sums and differences. Your answer is the multiplication of the moduli and the addition of the arguments: z1z2= r1r2[cos(theta 1 + theta 2) + i sin (theta 1 and theta 2)].
The product complex number is [tex]z_1z_2[/tex]= [tex]r_1r_2[/tex][cos (θ1+θ2) + i sin (θ1+θ2)].
What is Complex number?A real number and an imaginary number are effectively combined to create a complex number. The complex number is written as a+ib, where a and ib are real and imaginary numbers, respectively.
We have,
z1=r1 (cos θ1 +i sin θ1)
z2=r2 (cos θ2 +i sin θ2)
where r-radius/radii
and, m-modulus/moduli
Now, the product is
[tex]z_1z_2[/tex] = r1 (cos θ1 +i sin θ1) x r2 (cos θ2 +i sin θ2)
[tex]z_1z_2[/tex] = r1 r2(cos θ1 . cos θ2 +i cos θ1 sin θ2 + i sin θ1 cos θ2 + i² sin θ1 sin θ2)
[tex]z_1z_2[/tex] = r1 r2(cos θ1 . cos θ2 +i cos θ1 sin θ2 + i sin θ1 cos θ2 - sin θ1 sin θ2)
We know,
cos (a + b)= cos a cos b - sin a sin b
sin (a+b) = sin a cos b + cos a sin b
Thus, [tex]z_1z_2[/tex] = [tex]r_1r_2[/tex][cos (θ1+θ2) + i sin (θ1+θ2)].
Learn more about complex number here:
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#SPJ2
Find the number of permutations of the letters of the word LOOPHOLE
Answer:
6!/3!=6x5x4x3x2/3x2=6x5x4=120
Step-by-step explanation:
There are L, O, O, P, H, O, L, E; 8 letters in LOOPHOLE. However, three letters repeat: OOO. That means we cannot count them as different and use ! for them.
L,P,H,L,E,O are the letters not double counted. There are SIX letters, so 6!, however, you still have to divide since there are the Os you have to consider.
You have to divide by 3! as they repeat 3 times.
Thus, 6!/3!
"There is a subset of permutations that takes into account that there are double objects or repetitions in a permutation problem. In general, repetitions are taken care of by dividing the permutation by the factorial of the number of objects that are identical" -www. ck12. org > probability > lesson > permutation
How do I factor 4r^2 + 36r + 81
Answer:
How to solve your problem 42+36+814r2+36r+81
Use the sum-product pattern 42+36+814r2+36r+8142+18+18+81
barbara purchased stock last year for 8500 and paid a 1.25% broker fee. she sold it for 7324 and had to pay a .5% broker fee.
Barbara's net loss on the sale of the stocks she purchased last year is 1245.63.
What is the net loss?A net loss is earned when the cost price of purchasing the stock is greater than the selling price.
Net loss = cost price - selling price
Cost price = 8500 + (8500 x 1.25%) = 8606.25
Selling price = 7324 + (7324 x 0.5%) = 7360.62
Net loss = 8606.25 - 7360.62 = 1245.63
Here is the complete question: barbara purchased stock last year for 8500 and paid a 1.25% broker fee. she sold it for 7324 and had to pay a .5% broker fee. Compute her net loss.
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help if ur smart plssssssss
The answer would be ⁻[tex]\frac{19}{x-5\\}[/tex]
Answer:
[tex]\underline{\boxed{\sf \cfrac{-19}{x-5}}}[/tex]
Step-by-step explanation:
[tex]\sf \cfrac{x^2+10+25}{x+5}-\cfrac{x^2-6}{x-5}[/tex]
First, Let's factor the expressions that are not already factored in x^2+10+25/x+5
[tex]\sf \cfrac{(x+5)^2}{x+5}-\cfrac{x^2-6}{x-5}[/tex]
Now, cancel out x+5 in both numerator and denominator:-
[tex]\sf x+5-\cfrac{x^2-6}{x-5}[/tex]
Expand equations to make their denominators the same before adding or subtracting them. Multiply x+5 * x-5/x-5:-
[tex]\sf \cfrac{(x+5)(x-5)}{x-5}-\cfrac{x^2-6}{x-5}[/tex]
Now, Since [tex]\sf \frac{(x+5)(x-5)}{x-5}[/tex] and [tex]\sf \frac{x^2-6}{x-5}[/tex] and similar denominator, we'll subtrac their numerators.
[tex]\sf \cfrac{(x+5)(x-5)-(x^2-6)}{x-5}[/tex]
[tex]\sf \cfrac{x^2-5x+5x-25-x^2+6}{x-5}[/tex]
Combine like tems in [tex]\sf x^2-5x+5x-25-x^2+6[/tex] :-
[tex]\sf \cfrac{-19}{x-5}[/tex]
Therefore, your answer is D.
Please help me answer this
Answer:
d is the answerbecause a= 1,1 then 1+1 = 2
Which equation can be used to solve the problem below?
If four times a number is increased by 15, the result is three less than six times the number. Find the number.
Question 7 options:
4 (x + 15) = 6x -3
4x + 15 = 6(x-3)
4x + 15 = 6x -3
4x + 15 = 3-6x
Answer:
4x + 15 = 6x - 3
Step-by-step explanation:
Four times an unknown number is 4x, add 15 like it says, then it says three less so it would be - 3 and then it says than six times the [unknown] number, which remains being x
hope this helps!
3 more than the product of a number and six is equal to 2
6x + 3 = 2
its the answer
easy peasy
Answer:
3 + 6(x) = 2
Step-by-step explanation:
Given statement:
3 more than the product of a number and six is equal to 2
Converting the statement into an equation:
{3} {more} than {the product of a number and six} is {equal to} {2}
↓ ↓ ↓ ↓ ↓
3 + 6 × x = 2
⇒ Equation formed: 3 + (6 × x) = 2
⇒ Equation formed: 3 + 6(x) = 2
Which expression is equivalent to
(3^2) ^-2. -81 , -12 , 1/81 , 1/12
Answer:
1/81
Step-by-step explanation:
First lets simplfy the equation :
(3^2)^-2
(remember that you can mutiply the exponent outside the paranthesis by the exponent inside so -2*2=-4)
3^-4
Now lets evaluate this further!
3^-4
(negative exponets mean that you have to raise the number but as the denominator of a fraction)
1/3^4
Now we can solve it fully
1/3^4
1/81
there were 3 coins and they were tossed 200 times total. 23 times, 3 of them showed heads, 72 times, 2 of them showed heads, 77 times, 1 of them showed heads, 28 times, 0 showed heads. whats probability for each of the events?
Answer:
Step-by-step explanation:
If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.Solution:Probability of an event, P(E) = Number of occurrences where the event takes place / Total number of occurrencesThe probability of 2 heads coming up when three coins are tossed simultaneously is given by the ratio of the number of times a particular outcome occurs to the total number of tosses.Total number of tosses = 200Number of 2 heads outcomes = 72Probability of 2 heads outcomes = Number of 2 heads outcomes / Total number of tosses= 72/200= 9/25
How much time has passed? Start time: 1:18 a.m. End time: 3:09 a.m.
Answer:
1 hour and 51 minutes
Step-by-step explanation:
Find the difference between the time.
Assume that Quick Release lets go of the ball 6 feet above the ground and the receiver
catches it 6 feet above the ground. The ball reaches a maximum height of 16 feet above
the ground halfway to the receiver. Write an algebraic rule that models the path of the
football. Show all your work and explain your reasoning.
The equation that models the path of the football released from the given position is h(t) = -10 + 25.37t - 16.1t².
Initial velocity of the ball
The initial velocity of the ball is determined by applying third kinematic equation as shown below;
Vf² = V₀² - 2gh
at maximum height, the final velocity, Vf = 0maximum height reached from the point of projection, = 16 ft - 6ft = 10 ft0 = V₀² - 2gh
V₀² = 2gh
V₀ = √(2gh)
V₀ = √(2 x 32.17 x 10)
V₀ = 25.37 ft/s
Equation of motion of the ballThe equation of the ball's motion can be modelled as follows;
h = V₀t - ¹/₂gt²
10 = 25.37t - ¹/₂(32.17)t²
10 = 25.37t - 16.1t²
h(t) = -10 + 25.37t - 16.1t²
Thus, the equation that models the path of the football at any given position is h(t) = -10 + 25.37t - 16.1t².
Learn more about equation of motion here: https://brainly.com/question/25951773
What is the value of x in the figure below?
15
C
D
B
X
20
A.
20
15
B. 10
C. 300
D. 5
E 20
OF
F.
45
4
Answer:
choice f
Step-by-step explanation:
take ∆ abc
ab^2 = ac^2 + ab^2
400 = 225 + ab^2
ab =√ 175
∆aDB
bd = 20-x
ad^2 = 175 -(20-x)^2
∆acd
225 = x^2 + 175 - 400+ 40x - x^2
225 = -225 + 40x
450 = 40x
x = 45/4
could i have some super super quick help? :(( (part iv is not necessary)
might be getting the hang of it but i would still like some help
i)
Factor using completing square method
[tex]\dashrightarrow \sf y = -x^2+16x-64[/tex]
[tex]\dashrightarrow \sf y = -(x^2-16x+64)[/tex]
[tex]\dashrightarrow \sf y = -(x^2-8x-8x+64)[/tex]
[tex]\dashrightarrow \sf y = -(x(x-8)-8(x-8))[/tex]
[tex]\dashrightarrow \sf y = -((x-8)(x-8))[/tex]
ii)
Find zeros of a function, f(x) = 0
[tex]\dashrightarrow \sf -((x-8)(x-8))=0[/tex]
[tex]\dashrightarrow \sf (x-8)=0, \ (x-8)=0[/tex]
[tex]\dashrightarrow \sf x = 8[/tex]
iii)
In order to find vertex use the formulae : x = -b/2a
[tex]\dashrightarrow \sf x = \dfrac{-(16)}{2(-1)}[/tex]
[tex]\dashrightarrow \sf x = 8[/tex]
Then find y:
[tex]\dashrightarrow \sf y = -(8)^2+16(8)-64[/tex]
[tex]\dashrightarrow \sf y = 0[/tex]
coordinates: (8, 0)
iv) Sketched Below:
#1
y=-x²+8x+8x-64y=-x(x+8)+8(x+8)y=(x+8)(-x+8)y=-(x-8)(x-8)#2
Zeros are the x intercepts
Here they are
8,8#3
y=-x²+16x-64y=-[x²-16x+64]y=-(x-8)²+0Vertex form of parabola:-y=a(x-h)²+k
Vertex=(h,k)=(8,0)#4
Attached
the sum of the polynomials shown below
(10x2 + 3x - 6)
(2x2 9x - 12)
Answer:
[tex] \longrightarrow \sf \purple{ \boxed{ \bold{ \: 12x² + 12x – 18}}}[/tex]
Step-by-step explanation:
Simplify
10x² + 3x – 6 + 2x² + 9x – 12
Steps
1Ox² + 3x- 6 - 2x² + 9x - 12
Step 1:- Group like terms:
= 10x² + 2x²+ 3x + 9x - 6 - 12
Step 2:- Subtract the numbers: – 6–12 = -18
= 10x² + 2x² + 3x + 9x – 18
Step 3:- Add similar elements: 10x² + 2x² – 12x²
= 12x² + 3x + 9x – 18
Step 4:- Add similar elements: 3x + 9x – 12x
= 12x² + 12x – 18
If vector r = ❬4, 9❭ and s = ❬7, –5❭ then which is r – s?
r – s = ❬–3, 14❭
r – s = ❬3, –14❭
r – s = ❬–11, 4❭
r – s = ❬11, 4❭
Answer:
(a) <-3, 14>
Step-by-step explanation:
Addition and subtraction of vectors and matrices is done on a term-by-term basis.
__
The difference is ...
r - s
= <4, 9> -<7, -5>
= <4 -7, 9 -(-5)>
= <-3, 14> . . . . . . . matches the first choice
The difference between the squares of two numbers is 15. Three times the square of the first number increased by the square of the second number is 49. Find the numbers
Answer: 256,1
Step-by-step explanation:
[tex]\sqrt{x} - \sqrt{y} =15\\\sqrt{y} = 49-3\sqrt{x} \\\\\sqrt{x} -(49-3\sqrt{x} ) =15 \\so, x=256\\substitute \{x}\ :\\\\\sqrt{y} = 49-3\sqrt{256} =y=1[/tex]
so, 256=x
and, y=1
Hope this Helped!
Answer:
1, 4 or -1, -4
Step-by-step explanation:
Let the two numbers be x and such that x > y.
According to the first condition:
[tex]x^2 -y^2= 15[/tex]
[tex]\implies x^2 =y^2+15[/tex]......(1)
According to the second condition:
[tex]3x^2 +y^2= 49[/tex]
[tex]\implies 3(y^2+15) +y^2= 49[/tex]
(From equation 1)
[tex]\implies 3y^2+45 +y^2= 49[/tex]
[tex]\implies 4y^2 =49-45[/tex]
[tex]\implies 4y^2 =4[/tex]
[tex]\implies y^2 =\frac{4}{4}[/tex]
[tex]\implies y^2 =1[/tex]
[tex]\implies y =\pm 1[/tex]
When y = 1
[tex]\implies x^2 =(1)^2+15=1+15=16[/tex]
[tex]\implies x =\pm 4[/tex]
When y = -1
[tex]\implies x^2 =(-1)^2+15=1+15=16[/tex]
[tex]\implies x =\pm 4[/tex]
Thus, the required numbers are either 1, 4 or -1, -4
Help picture below problem 5
[tex]\angle F+ \angle E = 180^{\circ}\\\\ \implies \angle F + 164^{\circ} = 180^\circ\\\\\implies \angle F = 180^\circ - 164^\circ\\\\\implies \angle F= 16^\circ[/tex]
Solve the equation.
7 t − 3 = − 59
t=_____
Hey there!
7t - 3 = -59
ADD 3 to BOTH SIDES
7t - 3 + 3 = -59 + 3
CANCEL out: -3 + 3 because it give you 0
KEEP: -59 + 3 because it help solve for the t-value
NEW EQUATION: 7t = -59 + 3
SIMPLIFY IT!
7t = -56
DIVIDE 7 to BOTH SIDES
7t/7 = -56/7
CANCEL out: 7/7 because it give you 1
KEEP: -56/7 because it help solve for the t-value
NEW EQUATION: t = -56/7
SIMPLIFY IT!
t = -8
Therefore, your answer is: t = -8
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
Consider the following two quantities:
•Quantity A:
The slope of graphed line.
•Quantity B:
The y-value of the graphed line’s y-intercept.
Use the graph of the line K to decide which of the following statement is true?
a) Quantity A is greater
b) Quantity B is greater
c) The two quantities are equal
d) The relationship cannot be determined from the indicated information.
The slope of the line is -2.5 and the y-intercept is 2. This shows that the Quantity B is greater that A
How to find the intercept of a lineThe intercept is the point where the line intersects
Determine the slope of the graph using the coordinate points (0, 2) and (0.8, 0)
Slope = -2/0.8
Slope = -2.5
The slope of the line is -2.5 and the y-intercept is 2
This shows that the Quantity B is greater than A
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A (1,7)
B(8,-5)
distance formula
Answer:
The distance betweem the two points is about 14
Step-by-step explanation:
The distance formula is [tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]
According to the problem: [tex]x_{2}=8[/tex], [tex]x_{1} =1[/tex], [tex]y_{2} =-5[/tex], and [tex]y_{1} =7[/tex]
Substitute these numbers into the problem to get [tex]d=\sqrt{(8-1)^{2}+(-5-7)^{2}}[/tex]
[tex](8-1)^2=(7)^2=49[/tex]
[tex](-5-7)^2=(-12)^2=144[/tex]
[tex]\sqrt{49+144}=\sqrt{193} =13.8923349894[/tex]
You can round this number to approximately 14.
GIVING BRAINLIEST
What is the approximate distance between points A and B?
3.61
9.22
10.35
12.62
Answer:
9.22
I really hope it helps
let me know am i right to be shure
Answer:
[tex]9.22[/tex]
Step-by-step explanation:
You would need to use the distance formula to find the approximate distance between points A and B.
[tex]distance=\sqrt{(x_{2}-x_{1})^{2} +(y_{2}-y_{1})^2}[/tex]
[tex]\sqrt{((-2)-4)^{2} +((-4)-3)^2}[/tex]
[tex]\sqrt{(-6)^{2} +(-7)^2}\\[/tex]
[tex]\sqrt{36 +49}[/tex]
[tex]\sqrt{85}[/tex]
[tex]\sqrt{85}=9.219544457=9.22[/tex]
Therefore, the approximate distance between points A and B is 9.22.
Which angles are adjacent to each other? Select all that apply. 3 2 23 and 24 22 and 23 22 and 24 Z2 and 1
Answer:
angle 2 and angle 4
Simplify this expression.
7ײ+9ײ-ײ
Step-by-step explanation:
7x² + 9x² - x²
when you wrote the expression you used the "multiplication ×" and not "x".
anyway, when you have terms of the same variable of the same exponent, then the combination variable and exponent become an item. and the same items can simply be added in an expression.
this is similar to
7 apples + 9 apples - 1 apple
what would you do there ?
see ?
and the same we are doing now for the given expression : we are adding the terms up :
7x² + 9x² - x² = (7 + 9 - 1)x² = 15x²
and in the middle I gave you even the reason why we can do that.
please, let me know, if this is still unclear.
3. Every morning, Matthew fills his dog’s water dish with 16 oz of water. If his dog
finishes his water every day, how many ounces will his dog drink in a week?
How many cups is this?
How many pints is this?
How many quarts is this?
Drag each tile to the correct box.
Match the expression to the exponent property that you use first to simplify the expression.
Answer:
First box, second from the left,
Second box, first on the left.
Third box, third from the left.
Fourth box, Last on the right.
Step-by-step explanation:
Welcome.
could i have some help?
if you want another 50 you could answer the question i asked before this one lol
Answer:
Factoring out Greatest Common Factor (GCF)
[tex]6xy+12x^2y^2-4x^3y^3[/tex]
Factor out greatest common term [tex]2xy[/tex]:
[tex]\implies 2xy(3+6xy-2x^2y^2)[/tex]
Factoring by Grouping
[tex]20x^2+11x-3[/tex]
[tex]\implies a=20, b=11, c=-3[/tex]
[tex]\implies ac=20 \cdot -3=-60[/tex]
Find 2 two numbers that multiply to ac (-60) and sum to b (11)
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Therefore, the two numbers are 15 and -4
Rewrite b as the sum of these 2 numbers:
[tex]\implies 20x^2+15x-4x-3[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies 5x(4x+3)-1(4x+3)[/tex]
Factor out the common term (4x + 3):
[tex]\implies (5x-1)(4x+3)[/tex]
Factoring by Grouping
[tex]3x^2+3xa-2x-2a[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies 3x(x+a)-2(x+a)[/tex]
Factor out the common term (x + a):
[tex]\implies (3x-2)(x+a)[/tex]
Minerals and deposits are connected because
Step-by-step explanation:
This particular mineral was found in Finland. A mineral is a naturally occurring crystalline solid that cannot be physically broken down into smaller components. Deposits of minerals form when a medium that contains and transports mineral-making ore releases and deposits the ore.
PLEASE HELP!! ILL MARK AS BRAINLIEST AND GIVE 20 POINTS!!!!!
Select the correct answer.
What is the value of this expression when c= 25 and d= 14?
Answer: -10 or D
Step-by-step explanation: simplify radical then combine
3/5 x 4 as a fraction