Mathematics College

write the quotient in standard form:
8-7i/1-2i

Answers

Answer 1
Answer:
22/5+(9/5)i
To get rid of i in the denominator, use the difference-of-two-squares formula: (a+b)(a-b) = a²-b²

The denominator of your fraction is 1-2i. If we were to multiply the denominator by (1+2i), we would get (1-2i)(1+2i) = 1-(-4) = 5.

Of course, if we are going to multiply the denominator by (1+2i), we must also multiply the numerator by (1+2i). So your numerator becomes:

(8-7i) * (1+2i) = 8 + 16i - 7i +14 = 22+9i

Thus,

(8-7i) / (1-2i) = (22+9i) / 5

But standard form requires that we complete the division. So:

(8-7i) / (1-2i) = 22/5 + (9/5)i

The answer, then, is 22/5 + (9/5)i
Answer 2

The quotient in standard form for (8 - 7i) / (1 - 2i) is (22 + 9i) / 5. In this form, the numerator has real and imaginary components, while the denominator is a real number.

The division is simplified, providing a representation of the complex number in a standard format.

To write the quotient result of division in standard form,

Get rid of the imaginary unit 'i' from the denominator.

To do that, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 1 - 2i is 1 + 2i.

So, let's perform the multiplication:

(8 - 7i) / (1 - 2i) × (1 + 2i) / (1 + 2i)

Now, use FOIL (First, Outer, Inner, Last) method to expand the numerator:

Numerator = (8 - 7i)(1 + 2i)

= 8 + 16i - 7i - 14i² (remember that i² is -1)

= 8 + 9i - 14(-1)

= 8 + 9i + 14

= 22 + 9i

Now, let's expand the denominator:

Denominator = (1 - 2i)(1 + 2i)

= 1 + 2i - 2i - 4i²

= 1 - 4i²

= 1 - 4(-1)

= 1 + 4

= 5

Therefore, the quotient in standard form is (8 - 7i) / (1 - 2i) = (22 + 9i) / 5.

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Related Questions

which algebraic expression represents the "phrase the quotient of -8 and the sum of a number and three"

A -8/g+3
B 8/g+3
C -8+g/3
D -8/g +3

Answers

"the quotient of -8 and the sum of a number and three," written symbolically, comes out to 

           -8
--------------------------     This most closely resembles answer choice "A."
        n + 3

I have used "n" instead of "g" here.


The algebraic expression represents the given phrase is -8/(g+3). Therefore, option A is the correct answer.

The given phrase is "the quotient of -8 and the sum of a number and three".

We need to identify which algebraic expression represents the given phrase.

What is an algebraic expression?

Algebraic expressions are the mathematical statement that we get when operations such as addition, subtraction, multiplication, division, etc. are operated upon on variables and constants.

Let the unknown number be g.

Now, the quotient of -8 and the sum of a number and three=-8/(g+3)

The algebraic expression represents the given phrase is -8/(g+3). Therefore, option A is the correct answer.

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6-4. find p(x = 4) if x has a poisson distribution such that 3p(x = 1) = p(x = 2)

Answers

A random variable [tex]X[/tex] following a Poisson distribution with rate parameter [tex]\lambda[/tex] has PMF

[tex]f_X(x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex]

We're given that [tex]3\mathbb P(X=1)=\mathbb P(X=2)[/tex], so

[tex]\dfrac{3e^{-\lambda}\lambda^1}{1!}=\dfrac{e^{-\lambda}\lambda^2}{2!}[/tex]
[tex]\implies3\lambda=\dfrac{\lambda^2}2[/tex]
[tex]\implies\lambda^2-6\lambda=\lambda(\lambda-6)=0[/tex]

[tex]\lambda[/tex] must be positive, so we arrive at [tex]\lambda=6[/tex], which means

[tex]\mathbb P(X=4)=\dfrac{e^{-6}6^4}{4!}=\dfrac{54}{e^6}\approx0.1339[/tex]

An omelet station uses 3 eggs to make each omelet. What is the cost for the total number of eggs in a single omelet if a case of 30 dozen eggs has an as-purchase cost of 32.40?

Answers

I think it's 0.27 (3*32.40/360)

what is 5.62 rounded to?

use all place values.

Answers

it could be rounded to 6.0 or 10 anything 5 or above gets rounded up to the nearest tenth
The answer could be 6 if you are rounding up to the nearest unit. If you are rounding to the nearest 10, it would be 10.

Jesse has a piece of wood that is 8 feet long. he needs to cut pieces that are 7/8 of a foot long. how many pieces will he be able to make

Answers

He will be able to make 9.

The diagram shows a square of side 7 cm with two quadrants drawn inside. Find the area of the shaded region. (take pi= 22/7)

Answers

The area of the shaded region:
A = r ² π - a²
where: r = d/2 = 7√2 / 2
A = ( 7√2 / 2 )² · 22/7 - 7²
A = 49 · 2 / 4  · 22/7 - 49
A = 49/2 · 22/7 - 49
A = 7 · 11 - 49
A = 77 - 49
Answer: A (shaded) = 28 

Determine whether each of these sets is the power set of a set, where a and b are distinct elements.
a.∅
c.{∅,{a},{∅,a}}

Answers

a. [tex]\varnothing[/tex] cannot be the power set of any set. Consult Cantor's theorem, which says that the cardinality of the power set of any set (even the empty set) is strictly greater than the cardinality of the set.

(No part b?)

c. Also not the power set of any set, because any power set must have [tex]2^n[/tex] elements, where [tex]n[/tex] is the cardinality of the original set. The cardinality of this set is 3, but there is no integer [tex]n[/tex] such that [tex]2^n=3[/tex]. This set would be a power set if [tex]\{\varnothing\}[/tex] (that is, the set containing the empty set) were a member of it.

Javier had $305 in his bank account. His bank charges a fee of $7.50 each month that a balance is below $500. If he makes no other deposits or withdrawals, how much money is in Javier’s account at the end of three months?

Answers

7.50*3 = 22.50
305 - 22.50 = 282.50

Answer: [tex]\$282.5[/tex]

Step-by-step explanation:

Given : Javier had $305 in his bank account.

His bank charges a fee of $7.50 each month that a balance is below $500.

If he makes no other deposits or withdrawals, then the amount of deductions  in Javier’s account at the end of three months will be :-

[tex]\$7.50\times3=\$22.5[/tex]

Now, the amount of money in Javier’s account at the end of three months :-

[tex]\$305-\$22.5=\$282.5[/tex]

You need 1 1 4 114 cups of sugar to make 20 cookies. How many cups of sugar will you need to make 14 cookies?

Answers

[tex]\bf \begin{array}{ccll} \stackrel{cups}{sugar}&cookies\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 114&20\\ s&14 \end{array}\implies \cfrac{114}{s}=\cfrac{20}{14}\implies \cfrac{114\cdot 14}{20}=s[/tex]

Mary invests £12000 in a saving account. The account pays 1.5% compound interest per year. Work out the value of her investment after 2 years.

Answers

Final answer:

To find the value of Mary's investment after 2 years in a saving account with compound interest, we can use the compound interest formula. Substituting the given values into the formula gives us £12363.61.

Explanation:

To calculate the value of Mary's investment after 2 years, we can use the compound interest formula: A = P(1 + r/n)^(nt), where:

A is the final amountP is the principal amount (initial investment)r is the annual interest rate (as a decimal)n is the number of times the interest is compounded per yeart is the number of years

In this case, the principal amount is £12000, the annual interest rate is 1.5% (0.015 as a decimal), and the number of times the interest is compounded per year is 1. Plugging these values into the formula, we get:

A = 12000(1 + 0.015/1)^(1*2)

Simplifying this, we get:

A = 12000(1.015)^2

Calculating further, we get:

A ≈ 12000(1.030225) ≈ £12363.61

Therefore, the value of Mary's investment after 2 years is approximately £12363.61.

How many real solutions does the system have Y=-3x-3 Y=x^2-3x+5

Answers

Y = -3x-3 has one real solution.

Y= x^2 -3x + 5 has no real solutions because b^2 - 4ac, the discriminant, is less than zero.
To find the solutions to each equation, we'll first have to set Y to 0. We have:[tex]-3x-3 &=0\\ x^2-3x+5=0[/tex]

To obtain the solutions for both, you'll have to solve them for x. The first equation is linear, so obtaining a solution there is fairly straightforward, and you're guaranteed to get one real solution there. The second equation is a little more involved, and we'll need the quadratic formula to settle that one out. You might remember from earlier math classes that the quadratic formula gives you a way to quickly find the roots of a particular quadratic equation. Here's the full thing:

[tex]x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} [/tex]

For the sake of this question, the only part we're concerned with is the [tex]b^2-4ac[/tex] bit, which is referred to as the descriminant of the quadratic. If [tex]b^2-4ac \geq 1[/tex], we'll have two real solutions, since we'll need to evaluate the square root of the term for both positive and negative outcomes. If [tex]b^2-4ac=0[/tex], we only have one real solution, since adding and subtracting 0 from the [tex]-b[/tex] term have the same effect. If [tex]b^2-4ac \ \textless \ 0[/tex], we have no real solutions; the discriminant is nested inside a square root, and taking the square root of a negative number only produces imaginary results.

With that in mind, let's look at the discriminant of [tex]x^2-3x+5[/tex].

Looking at the coefficients and constant, we have a=1, b=-3, and c=5, which makes our discriminant

[tex](-3)^2-4(1)(5) = 9-20=-11[/tex]

-11 is less than 0, so we have no real solutions to the second equation. This means that our first equation is the only one with a real solution, so the total number of real solutions for the system is 1.

Present 5 mixed numbers that add up to the whole number 10.

Answers

5 1/3 + 4 2/3 = 10
4 1/3 + 5 2/3 = 10
6 1/4 + 3 3/4 = 10
9 3/5 + 2/5 = 10
2 8/10 + 7 1/5 = 10

hope this helps

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solve this system of linear equations. separate the x- and y- values with a coma. 9x=27-9y
20x=71-9y

Answers

Solve the system

9x = 27-9y
20x=71-9y

Let's use the "elimination through addition/subtraction method.
Multiply the first equation by -1 so as to obtain +9y:

-9x = -27 + 9y
20x=  71-   9y
--------------------

Add these 2 equations together:

-9x = -27 + 9y
20x=  71-   9y
--------------------
 11x = 44

Solve this for x:  x = 44/11 = 4.

Now find y by subst. 4 for x in either of the original equations.

Using the second equation:     

20x=71-9y
20(4) = 71 - 9y
80-71 = -9y
9 = -9y.  Then y = -1.

The solution set is (4,-1).

sum of the interior angle measures of the polygon with 9 sides

Answers

Hello There!

The sum of interior angles for any polygon is:
(n - 2) x 180
Where 'n' is the number of sides.
In this case, the polygon is a nonagon, which has 9 sides.
Substitute the value in:
(9 - 2) x 180 = ?
Solve:
Sum of interior angles = 1260°.

Hope This Helps You!
Good Luck :) 

- Hannah ❤

Answer: 1260

Step-by-step explanation:

The formula f=5280m can be used to convert a measurement of f feet to an equivalent measurement m miles. If an airplane flies at 21,120 feet above the ground, how much is that in miles?

A. 4 miles
B. 5.3 miles
C. 111, 513, 600 miles
D. 132, 620, 600 miles

Answers

the answer is a
21,120/5280=4

Evaluate the line integral c y3 ds,
c.x = t3, y = t, 0 ≤ t ≤ 3

Answers

[tex]\displaystyle\int_{\mathcal C}y^3\,\mathrm dS=\int_{t=0}^{t=3}t^3\sqrt{1^2+(3t^2)^2}\,\mathrm dt=\int_0^3t^3\sqrt{1+9t^4}\,\mathrm dt[/tex]

Take [tex]u=1+9t^4[/tex] so that [tex]\mathrm du=36t^3\,\mathrm dt[/tex]. Then

[tex]\displaystyle\int_{\mathcal C}y^3\,\mathrm dS=\frac1{36}\int_{u=1}^{u=730}\sqrt u\,\mathrm du=\frac{730^{3/2}-1}{54}[/tex]

can someone please help with transformations of parent functions

Answers

[tex]\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\\\ % left side templates \begin{array}{llll} f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}(\mathbb{R})^{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}} sin\left({{ B }}x+{{ C}} \right)+{{ D}} \end{array}\\\\ --------------------[/tex]

[tex]\bf \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }{{ B}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the y-axis}[/tex]

[tex]\bf \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by }{{ D}}\\ \left. \qquad \right. if\ {{ D}}\textit{ is negative, downwards}\\\\ \left. \qquad \right. if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}}[/tex]

that said, compressing vertical |x| by  a factor of 4, means A = 4, thus is 4|x|.

Sally's soccer team won 68% of the games they played.if they won 17 games, how many did they play?

Answers

Let the number of games played be x.  68% of x comes out to 17 games.

Then 0.68x = 17.  Mult. both sides by 100:  68x = 1700.

Divide 68 into 1700 to obtain x, the number of games played:  x = 25

The team plays in 25 games.

Answer: The total number of games played are 25

Step-by-step explanation:

Let the total number of games played be 'x'

We are given:

Number of games won = 17

Percentage of games won = 68 %

Calculating the total number of games:

[tex]68\% \text{ of x}=17[/tex]

So,

[tex]\frac{68}{100}\times x=17\\\\x=\frac{17\times 100}{68}=25[/tex]

Hence, the total number of games played are 25

The sale price of an item is $45 after a 40% discount is applied. What is the original price of the item?

Answers

Diving the sales price by 60% (100-40) will obtain the original price. 
$45/60% = $75 was the original sales price. 

The original price of the item is $75.

Suppose the original price of the item =x

Discount =40%

What is the percentage?

The percentage is the value per hundred.

Given discounted price or sale price of the item =$45

This means [tex]x-\frac{x*40}{100} =45[/tex]

[tex]\frac{60x}{100} =45[/tex]

[tex]\frac{3x}{5} =45[/tex]

[tex]x=75[/tex]

So, the original price of the item =$75

Hence, the original price of the item is $75.

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What value of x is in the solution set of 2(3x – 1) ≥ 4x – 6?

Answers

It can be convenient to subtract the right side of the inequality from both sides.

... 2(3x -1) -(4x -6) ≥ 0

... 2x +4 ≥ 0 . . . . . . . . . collect terms

... x +2 ≥ 0 . . . . . . . . . . divide by the coefficient of x

... x ≥ -2 . . . . . . . . . . . . add the opposite of the constant

Any value of x that is at least -2 will be in the solution set.

Please help me to Evaluate P(6,2)

Answers

The value of the permutation [tex]_6P_2[/tex] equals 30.

The notation P(n,r) represents the number of permutations of r elements taken from a set of n distinct elements.

The formula for permutations is expressed as:

[tex]P(n,r) = \frac{n!}{(n-r)!}[/tex]

Given the permutations:

[tex]_6P_2[/tex]

n = 6 and n = 2:

Plug these into the above formula:

[tex]P(n,r) = \frac{n!}{(n-r)!}\\\\P(n,r) = \frac{6!}{(6-2)!}\\\\P(n,r) = \frac{6!}{4!}\\\\P(n,r) = \frac{6*5*4*3*2*1}{4*3*2*1}\\\\P(n,r) = \frac{6*5}{1}\\\\P(n,r) = 30[/tex]

Therefore, the value of [tex]_6P_2[/tex] is 30.

The sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. suppose the number of tickets written per day follows a poisson distribution with a mean of 6.5 tickets per day. interpret the value of the mean.

Answers

The Poisson distribution defines the probability of k discrete and independent events occurring in a given time interval.
If λ = the average number of event occurring within the given interval, then
[tex]P(k \, events) = e^{-\lambda } ( \frac{\lambda ^{k}}{k!} )[/tex]

For the given problem,
λ = 6.5, average number of tickets per day.
k = 6, the required number of tickets per day
The Poisson distribution is
[tex]P(k \, tickets/day)=e^{-6.5} ( \frac{6.5 ^{k}}{k!} )[/tex]
The distribution is graphed as shown below.

Answer: 
The mean is λ = 6.5 tickets per day, and it represents the expected number of tickets written per day.
The required value of k = 6 is less than the expected value, therefore the department's revenue target is met on an average basis.

Complete this item.

For the following figure, can you conclude that l | | m? Select true or false.

Answers

False because that’s unlegibel

Answer: False. Both angles are not equal therefore the lines l and m are not parallel.

Determine which of these relations are transitive. the variables x, y, x', y' represent integers.

Answers

Answer: A. xâĽy if and only if xy is positive. B. xâĽy if and only if x+y is even. C. xâĽy if and only if xâ’y is a multiple of 10. D. xâĽy if and only if xâ’y is positive. E. xâĽy if and only if x+y is odd. A. xy is positive, so xy>0. So if x >=1, then (x,x) is satisfied. True B. If x+y is even, then x and y are both even or both odd. So odd(1,1) is even and even(2,2) is even. Therefore true C. x-y is a multiple of 10, but x-x is always 0, so its false. D. x-y is positve, but if x(1) and y(0) then (x,x) is 0, not positive, therefore false. E. x+y is odd therefore x is odd and y is even, or x is even and y is odd. But even + even, and odd + odd is always even. Therefore false.

Find the probability that in a family of 4 children there will be
a.at least 1 boy
b.at least 1 boy and at least 1 girl
c.out of 2000 families with 4 children each how many would you expect to have 1 or 2 girls. assume that the probability of a male birth is 1/2.

Answers

It will be letter for family of 4 is letter c

Tickets for rides at the fair cost $0.75 each you can also buy a roll of 25 tickets for $15 what is the cost of each ticket if you buy a roll of tickets

Answers

You get 25 tickets for $15

all you need to do is take the cost and divide it by the number of tickets.

$15 ÷ 25 tickets = $0.60 per ticket

Final answer:

The cost per ticket when buying a roll of 25 tickets for $15 at the fair is $0.60, which is less than the single ticket price of $0.75.

Explanation:

The student has asked about the cost of each ticket if bought in a roll at a fair. To find the cost per ticket when purchasing a roll of 25 tickets for $15, you simply divide the total cost of the roll by the number of tickets in the roll.

Cost per ticket = Total cost of the roll / Number of tickets in the roll

Cost per ticket = $15 / 25

Cost per ticket = $0.60

Therefore, when buying a roll of tickets, the cost of each ticket is $0.60, which offers a saving compared to the single ticket price of $0.75.

What is the unit rate of 1,700 in 40 minutes

Answers

the unit rate of 1700/40 would be 42.5 hope this helped:)

Find a pair of 3 digit numbers that have an estimated difference of 520

Answers

The three digit number is such that it takes the general form of,
     100x + 10y + z
where x is the hundreds digit
           y is the tens digit, and 
           z is the ones digit

If we have one of the three digit numbers as 100 (being the smallest), we determine the second number by adding 520 to 100 which gives us an answer of,
       620

Therefore, one pair of these numbers is 620 and 100. 

To find a pair of 3 digit numbers with an estimated difference of 520, we can start with any 3 digit number and add or subtract 520. The pair 1000 and 480, or 950 and 430, for example, both have an estimated difference of 520.

To find a pair of 3 digit numbers that have an estimated difference of 520, you can choose any 3 digit number and then either add or subtract approximately 520 to find the second number. For instance, if you choose 1000 as a starting number, you could subtract 520 to get 480. Thus, the pair of numbers 1000 and 480 have an estimated difference of 520. It is important to choose numbers that when rounded to the nearest hundred, give an exact difference of 520. For example, 950 and 430 is another valid pair since 950-430=520. Remember that the difference is the result of subtracting the smaller number from the larger one.

Since Jenna's heart rate is 60 beats per minute, if her heart has beat 604,800 times, how many days since she was born?

Answers

seven days since she was born
Final answer:

By calculating the number of heartbeats in a day (based on the heart rate of 60 bpm) and dividing the total number of heartbeats by that value, we can estimate that Jenna is 7 days old.

Explanation:

The subject of this question is Mathematics, and it's suitable for high school grade level. To find out the number of days Jenna has lived using her heartbeats, you first need to calculate how many beats her heart makes in one day. Knowing that her heart rate is 60 beats per minute, we can calculate the number of beats in one hour by multiplying by 60 (number of minutes in one hour). So in one hour, we get 60 * 60 = 3600 beats. In 24 hours (one day), the heart will beat 3600 * 24 = 86,400 times. So, if Jenna's heart has beat 604,800 times in her life, we divide that number by the number of beats in one day to find out the number of days: 604,800 / 86,400 = 7 days. Therefore, Jenna is 7 days old.

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A direct variation function contains the points (2, 14) and (4, 28). Which equation represents the function?

Answers

a direct variation function, is just another way to word a "linear equation", and the slope will then be the "constant of variation".

so, in short, what's the equation of the line that runs through 2,14 and 4,28?

[tex]\bf \qquad \qquad \textit{direct proportional variation}\\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 2}}\quad ,&{{ 14}})\quad % (c,d) &({{ 4}}\quad ,&{{ 28}}) \end{array}[/tex]

[tex]\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{28-14}{4-2}\implies \cfrac{14}{2}\implies 7 \\\\\\ % point-slope intercept \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})}\implies y-14=7(x-2) \\\\\\ y-14=7x-14\implies y=7x[/tex]

we know that

A relationship between two variables, x, and y, represent a direct variation if it can be expressed in the form [tex]\frac{y}{x}=k[/tex] or [tex]y=kx[/tex]

In this problem we have

Points [tex](2,14)[/tex] and [tex](4,28)[/tex]

so

Find the value of k

First point

[tex]\frac{14}{2}=k[/tex]

[tex]k=7[/tex]

Second point

[tex]\frac{28}{4}=k[/tex]

[tex]k=7[/tex]

the function is

[tex]y=kx[/tex] --------> substitute the value of k

[tex]y=7x[/tex]

therefore

the answer is

[tex]y=7x[/tex]

In this problem with a single point was sufficient to calculate the equation, since in a direct variation the line passes through the origin, it is not necessary to use the formula of the slope


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