a rectangle has a length of 5^3 units and a width of 5^2 units what is the area of the rectangle.

[tex]\omega[/tex] is a 2020th root of unity, so [tex]\omega^{2020}=1[/tex] and we have a kind of reflection identity of [tex]\omega^{2020-n} = \omega^{-n}[/tex] for [tex]n\in\Bbb Z[/tex].

Let's evaluate the product first. Denote it by [tex]P_k[/tex]. We split the product where the [tex]j=k[/tex] factor would belong, and pull out powers of [tex]\omega^k[/tex].

[tex]\displaystyle P_k = \prod_{j=1, j\neq k} (\omega^k - \omega^j) \\\\ ~~~~ = \prod_{a=1}^{k-1} (\omega^k - \omega^a) \prod_{b=k+1}^{2019} (\omega^k - \omega^b) \\\\ ~~~~ = (\omega^k)^{k-1} \prod_{a=1}^{k-1} (1-\omega^{a-k}) \cdot (\omega^k)^{2019-k} \prod_{b=k+1}^{2019} (1 - \omega^{b-k}) \\\\ ~~~~ = (\omega^k)^{2018} \prod_{a=1}^{k-1} (1 - \omega^{-a}) \prod_{b=1}^{2019-k} (1 - \omega^b) \\\\ ~~~~ = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \prod_{b=1}^{2019-k} (1-\omega^b)[/tex]

Now introduce some factors to "complete" the [tex]b[/tex]-product and have it contain 2019 factors.

[tex]\displaystyle P_k = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \frac{\displaystyle \prod_{b=1}^{2019} (1 - \omega^b)}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}[/tex]

It's relatively straightforward to show that if [tex]\zeta[/tex] is an [tex]n[/tex]-th root of unity, then

[tex]\displaystyle \sum_{m=1}^{n-1} (1-\zeta^m) = n[/tex]

which gives

[tex]\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}[/tex]

Shifting the index in the denominator and again using the reflection property eliminates all but one factor.

[tex]\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=1}^k (1 - \omega^{-b})} \\\\ ~~~~ = \frac{2020 \omega^{-2k}}{1 - \omega^{-k}}[/tex]

Now evaluate the sum. We can exploit symmetry. Split the sum at the 1010th term, so that

[tex]\displaystyle - \sum_{k=1}^{2019} P_k = -2020 \left(\sum_{k=1}^{1009} \frac{\omega^{-2k}}{1 - \omega^{-k}} + \frac{\omega^{-2020}}{1-\omega^{-1010}} + \sum_{k=1011}^{2019} \frac{\omega^{-2k}}{1-\omega^{-k}}\right)[/tex]

The middle terms reduces to 1/2. Shifting the index in the second sum, we can condense it to

[tex]\displaystyle -\sum_{k=1}^{2019} P_k = -2020 \left(\frac12 + \sum_{k=1}^{1009} \left(\frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k}\right)\right)[/tex]

Join the fractions.

[tex]\displaystyle \frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k} = 1 - \frac{2-\omega^{2k}-\omega^{-2k}}{2-\omega^k-\omega^{-k}} = -(1+\omega^k + \omega^{-k})[/tex]

The remaining sums are trivial.

[tex]-\displaystyle \sum_{k=1}^{2019} P_k = -2020 \left(\frac12 - 1009 - \frac{1-\omega^{1010}}{1-\omega} - \frac{1-(-\omega)^{1010}}{1+\omega}\right) \\\\ ~~~~ = 2020\cdot1009 - 1010 \\\\ ~~~~ = (2000+20)(1000+9) - 1000 - 10 \\\\ ~~~~ = 2\cdot1000^2 + 37\cdot1000 + 170[/tex]

Taking this last result mod 1000, we find the last 3 digits to be 170.

Answer:

4 × 10^6

Because 10^6 = 100,000

Answer:

4×10^6

Step-by-step explanation:

you write a non zero digit which is 4 placing your decimal after the first non zero digit ..then you count the no. of digits you need to move the beginning decimal to get to where your decimal is ...or to use a scientific calculator to make it simple

The nonlinear- function which represents b(9)=4 is given by b(x)=72/x - 4 .

In mathematics and physics, a nonlinear system is one in which the change in the output is not proportional to the change in the input. Engineers, biologists, physicists, mathematicians, and many other scientists are interested in nonlinear problems since the majority of systems are inherently nonlinear.

Nonlinear function graphs don't resemble lines at all. It contains the formula f(x) = ax + b. Its equation can take any form, with the exception of f(x) = ax + b. The slope of the curve is the same between any two points. The point pairs on the graph do not all have equal slopes.

Of all the given options: the only equations which imply that b(9)=4  are :

i) b(x)=72/x - 4 and  (iv) b(x)=4

Of this two options only the first option is non-linear.

Hence the required option is  b(x)=72/x - 4 .

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The missing options are:

i)   b(x) = 72/x - 4

ii) b(x) = √x + 7

iii)b(x)  = 2x+1

iv) b(x) = 4

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Answer:

Q1 = 8

Hope this helps with your question :)

The line at the start of the "box" is Q1, the middle line in the "box" is the median also called Q2, and the last line of the "box" is Q3.

According to the solving the simplified value of the equation 1/(x^4)^-2 is

= x⁸.

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction and is the main rule for simplifying expressions.

Simplification seems to be the process of substituting a mathematical expression with a simpler, usually shorter counterpart.

Designers employ simplification to make designs more memorable and instantly recognizable in logos, emblems, posters, computer icons, as well as other design applications.

Given :[tex]\frac{1}{\left(x^4\right)^{-2}}[/tex]

On simplifying we get:

Use  (aⁿ)ˣ  = aⁿˣ

so we get:

[tex]\frac{1}{X^{(4)(-2)}}[/tex]

=[tex]\frac{1}{x^{-8}}[/tex]

Use a⁻ⁿ = 1/aⁿ

Now:

= [tex]\frac{1}{\frac{1}{x^8}}\\=x^8[/tex]

According to the solving the simplified value of the equation 1/(x^4)^-2 is

= x⁸

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Answer:

y = -2x + 10

Step-by-step explanation:

An addition equation for calculating volume is = 12 units cube.

The fundamental addition formula, sometimes known as the mathematical equation of addition, is as follows. The addends are 5 and 3, and the sum is 8. It should be noted that an addition fact can have many addends. For instance, 5 + 7 + 9 + 3 = 24.

Volume refers to the amount of space occupied by a 3D shape. The volume of a shape can be calculated by multiplying its height, breadth, and depth. If somehow the shape is made using cubic cm blocks, the volume can be calculated by counting the cubes.

Total no. of cubes  = 12

length of the one cube = 1 unit

So,

volume of the given prism = l x b x h

volume of one prism = 1 unit^3

There are 12 cube are present on adding all the units we get

volume of total cubes are  = 12 unit cube.

So,

The volume is  = 12 unit cube.

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Answer:

-1

Step-by-step explanation:

make it equal to zero

-9y - 9 = 0

add 9 to both sides

-9y = 9

divide both sides by -9 to get y alone

y = 9/-9

simplify

y = -1

Answer:30

Step-by-step explanation:because 30 is a multiple and factor of 30.

The cost of gas in England is 96.4 pound sterling/liter is 549.1185 U.S. dollars/gallon

The cost of gas in England is 96.4 pound sterling/liter.

3.875 liters = 1 gallon

1 liter = 1/3.875 gallon

1.47 US$= 1 pound sterling

pound sterling / liter = 1.47/(1/3.875)) U.S. dollars/gallon

                                   = (1.47)(3.875) U.S. dollars/gallon

96.4 pound sterling / liter = (96.4)(1.47)(3.875) U.S. dollars/gallon

                                         = 549.1185 U.S. dollars/gallon

Sterling is the arena's oldest forex that is still in use and that has been in continuous use due to the fact that its inception.

it is presently the fourth maximum-traded currency in the forex market, after the USA dollar, the euro, and the Javanese yen.

collectively with the ones 3 currencies and Renminbi, it paperwork the basket of currencies which calculate the price of IMF unique drawing rights. As of mid-2021, sterling is likewise the fourth most-held reserve forex in international reserves.

these kinds of currencies are government-issued fiat currencies.

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Answer:

Variance = 25

Step-by-step explanation:

I'm smart

Answer: Devon would have -6 dollars left

Step-by-step explanation:

Answer:

Step-by-step explanation: C

The number of vehicle administered by department A only = 18

The number of vehicle administered by department B only = 8

The number of vehicle administered by department C only = 1

Given data

50 vehicles belong to an enterprise that has three departments, A, B and C

6 vehicles are available for the use of all three departments

2 vehicles give service to neither of the departments

24 vehicles are administrated by department B

16 of the 24 used by B are also used by the others departments

There are twice as many vehicles in department A as there are in department C.

equal numbers of vehicles are allocated when used by any two departments

number of vehicles on department A = 2 * number of vehicles in department C.

say A = 2C

B = 24

let x represent the number of vehicles used by any two departments

16 of the 24 used by B are also used by the others departments and equal numbers of vehicles are allocated when used by any two departments hence

16 = 2x + 6

2x = 16 - 6

2x = 10

x = 5

B only = 24 - 2x - 6 = 18 - 2x = 18 - 10 = 8

A only = A - 2x - 6 = 2C - 10 - 6 = 2C - 16

C only = C - 2x - 6 = C - 10 - 6 = C - 16

adding up all the vehicles as allocated to get to the sum of 50

24 - 2x - 6 + A - 2x - 6 + C - 2x - 6 + x + x + x + 6 + 2 = 50

24 - 6 - 6 - 6 + 6 + 2 - 2x - 2x - 2x + x + x + x + A + C = 50

24 - 10 - 6x + 3x + 2C + C = 50

14 - 3x + 3C = 50

3C - 3X = 50 - 14

3C - 3(5) = 36

3C - 15 = 36

3C = 36 + 15 = 51

C = 17

C only

C only = C - 16

C only = 17 - 16

C only = 1

A only
A only = 2C - 16

A only = 2 * 17 - 16 = 34 - 16 = 18

The number of vehicle administered by department A only = 18

The number of vehicle administered by department B only = 8

The number of vehicle administered by department C only = 1

The number of vehicle administered by department A = 2C = 34

The number of vehicle administered by department B = 24

The number of vehicle administered by department C = 17

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The values of x, y, and z. The diagram is not to scale.

This is further explained below.

Generally, Angle in a triangle is 180^{\circ}

Therefore

45+68+x=180

x=180-(45+68)

x=67

The angle on a line is also 180^{\circ}

x+z =180

67+z =180

z =180-67

z =113

In conclusion

180=113+12+y

-180-(113+12)=y

y=55

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A n B = {2,4,6,8}

The intersection of Sets :

The intersection of sets for two given sets is the set that contains all the elements that are common to both sets. The symbol for the intersection of sets is "∩''. For any two sets A and B, the intersection, A ∩ B (read as A intersection B) lists all the elements that are present in both sets, and are the common elements of A and B.

Intersections of sets have properties similar to the properties of numbers. The properties of the intersection of sets include the commutative law, associative law, the law of null set and universal set, and the idempotent law. The following table lists the properties of the intersection of sets.

Name of Property/Law Rule

Commutative Law A ∩ B = B ∩ A

Associative Law (A ∩ B) ∩ C = A ∩ (B ∩ C)

Law of ϕ and U ϕ ∩ A = ϕ , U ∩ A= A

Idempotent Law (A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Important Notes:

(A ∩ B) is the set of all the elements that are common to both sets A and B.

If A ∩ B = ϕ, then A and B are called disjoint sets.

n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

Given :

A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

B = {2, 4, 6, 8, 10, 12}

Find :

A n B = {2,4,6,8}

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Answer:

Are there options you forgot to put on the question?

Step-by-step explanation:

Answer

.9780

Step-by-step explanation:

Probability, not all 3 people are from Arizona

Arizona Equals 1-   2/91 =    89/91  = .9780

Probability is .9780

Based on the given parameters, the value of y in the figure is 13

The measures of the angles are given as:

Angle 1 = 8y

Angle 2 = 6y - 2

The angles are supplementary angles

So. we have

8y + 6y - 2 = 180

Evaluate the like terms

14y =182

Divide both sides by 14

y = 13

Hence, the value of y in the figure is 13

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Answer:

Step-by-step explanation:

You want the cost and the number of months of membership such that the cost of both gyms is the same.

The total cost of x months of membership at each gym will be the sum of its initial fee and the product of the monthly fee and the number of months.

The cost at Community Gym is ...

  c = 60 + 65x

The cost at Workout Gym is ...

  w = 190 +55x

These costs will be the same when ...

  c = w

  60 +65x = 190 +55x

Subtracting 60+55x from both sides of the equation leaves ...

  10x = 130

  x = 13 . . . . . . divide by 10

The cost will be the same after 13 months.

That cost will be ...

  60 +65(13) = 60 +845 = 905

The amount paid at each gym after 13 months will be $905.

Cary's and Dan's present ages are 44 and 35 years respectively.

Let the Dan's present age be x years.

According to the given question.

Cary is 9 years older then Dan.

And, after seven years the sum of their ages will be 93.

So, from the given conditon we can say that

cary's age = 9 + x

And after seven years, sum of the ages of Cary and Dan

9 + x + 7 + x + 7 = 93

⇒ 23 + 2x = 93

⇒ 2x = 93 - 23

⇒ 2x = 70

⇒ x = 70/2

⇒ x = 35

⇒ Dan's present age is 35 years.

Therefore, Cary's age = 35 + 9 = 44 years.

Hence, Cary's and Dan's present ages are 44 and 35 years respectively.

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Answer:

Step-by-step explanation:

the answer is 23.

Answer:

C

Step-by-step explanation:

Answer:

Step-by-step explanation: Let x be the smaller and y be the largest number.

Since x+y=13, we deduce y=13-x

Now, for the translation: "two more than the larger number" is y+2 , while "twice the smaller" is 2x

Their sum is y+2+2x

And since we know that y=13-x, we have y+2+2x=13-x+2+2=15+x

Answer:

x-6

Step-by-step explanation:

Answer:

y=3x+13

Step-by-step explanation:

Simply add the 3x to the other side.

Answer: y = 13 + 3x

Step-by-step explanation:

you just add 3x to both sides of the equation !! sorry if it doesn’t help but you basically put it the other way around and put 3x on both sides

Step-by-step explanation:

Here,

angle B = 3 x angle A

angle C = 6 x angle A - 16

now, for a triangle,

angle A + angle B + angle C = 180

or, angle A + 3 x angle A + 6 x angle A - 16 = 180

or, 10 x angle A = 196

so, angle A = 19.6 degrees

angle B = 3 x 19.6 = 58.8 degrees

angle C = 6 x 19.6 - 16 = 101.6 degrees

Answer:

its 63 you got it right

Step-by-step explanation: