Xander collected four times as many stamps as his cousin.If xander collected 60 stamps,then how many did his cousin collect What is the variable and the Equation

Answer:

Step-by-step explanation:

 102+202+302+402+502+602+702+802+902(4518)

+112+212+312+...+ 812+912(4608)

+122+222+322+...+822+922(4698)

+132+232+332+...+932(4788)

..........................................

+192+292+392+...+992(5328)

4518+4608+4698+...+5328

n=10

[tex]s=\frac{10}{2}(4518+5328)\\=5(9846)\\=49230[/tex]

Final answer:

To find the sum of all positive 3-digit numbers ending in 2, we calculate the total for each digit's place and sum them up, resulting in a total sum of 8280.

Explanation:

The problem requires finding the sum of all positive 3-digit numbers with a last digit of 2. To calculate this, we can identify that the first such number is 102 and the last is 992. There are 90 such numbers because they correspond to the tens digit going from 0 to 9 for each of the nine possible hundreds digits (1-9).

Since each number ends in 2, we can think of them as (100x + 10y + 2), where x is the hundreds digit (1 through 9) and y is the tens digit (0 through 9). To find the sum, we calculate the sum of the hundreds digits times their frequency, the sum of the tens digits times their frequency, and add 2 times the number of terms (90). The formula would be:

Sum = (Sum of hundreds values) * 10 * 9 + (Sum of tens values) * 1 * 90 + 2 * 90

The hundreds values are 1 through 9, whose sum is 45, and the tens values are 0 through 9, whose sum is 45 as well. Plugging these values into the formula, we get:

Sum = 45 * 10 * 9 + 45 * 1 * 90 + 2 * 90 = 4050 + 4050 + 180 = 8280.

Answer:

The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]

Step-by-step explanation:

Given:

Perimeter of new garden = 70 feet.

Let the length of the new garden be 'l'.

Also Let the width of the new garden be 'w'.

We need to find the dimension of new garden.

Now Given:

Length is 5 feet shorter than three times it width.

framing the equation we get;

[tex]l =3w-5 \ \ \ \ equation\ 1[/tex]

Now we know that;

Perimeter of rectangle is equal to twice the sum of length and width.

framing in equation form we get;

[tex]2(l+w)=70[/tex]

Now Diving both side by 2 using Division property of equality we get;

[tex]\frac{2(l+w)}2=\frac{70}{2}\\\\l+w =35[/tex]

Now Substituting equation 1 in above equation we get;

[tex]3w-5+w=35\\\\4w-5=35[/tex]

Adding both side by 5 Using Addition Property of equality we get'

[tex]4w-5+5=35+5\\\\4w=40[/tex]

Now Diving both side by 4 using Division property of equality we get;

[tex]\frac{4w}{4}=\frac{40}{4}\\\\w=10\ ft[/tex]

Now Substituting the value of 'w' in equation 1 we get;

[tex]l =3w-5\\\\l =3\times10-5\\\\l = 30-5\\\\l= 25\ ft[/tex]

Hence The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]

Answer:

whole numbers

Step-by-step explanation:

The domain is the number of people. The smallest number of people you could have would be 0 people so the appropriate domain is whole numbers.

Make me the brainliest

Answer:

whole numbers

Step-by-step explanation:

Answer:

Step-by-step explanation:

Total copies of books ordered by Lena for his book club members is 12. The cost of the book is $19 each. Since the books are the same, the total cost of the books will be

19 × 12 = $228

the order has a $15 shipping charge. It means that the total amount that Lena would pay for the 12 books is total cost of the books + shipping fee. it becomes

228 + 15 =

=$243

Answer: the measure of the indicated angle is 100 degrees

Step-by-step explanation:

The sum of angles in a triangle is 180 degrees. Let x represent the unknown angle in the bigger triangle. Therefore,

x + 80 + 25 = 180 degrees

x + 105 = 180

x = 180 - 105 = 75 degrees.

Let z represent the other unknown angle in the smaller triangle. Since the sum of the angles on a straight line is 180 degrees, therefore

75 + 55 + z = 180

130 +z = 180

z = 180 - 130 = 50 degrees

Let y represent the unknown angle that we are looking for. Therefore,

50 + y + 30 = 180

80 + y = 180

y = 180 - 80 = 100 degrees

Answer:

55

Step-by-step explanation:

Answer:

Number of statues that can be painted are 17

Step-by-step explanation:

Initially Diana has [tex]\frac{7}{8}[/tex] liters of paint remaining.

Every statue requires [tex]\frac{1}{20}[/tex] liters of paint for painting.

We have to find how many statues we will be able to paint with this remaining paint.

To get the number of statues,

Number of statues = [tex]\frac{Paint remaining}{Paint required for 1 statue}[/tex]

number of statues = [tex]\frac{\frac{7}{8} }{\frac{1}{20} }[/tex]

                               = [tex]\frac{35}{2}[/tex] = 17.5

Since the number of statues is not an integer the maximum number of statues that can be painted are 17.

Answer: 35/2

Step-by-step explanation:

Answer:

Step-by-step explanation:

The attached photo shows the diagram of quadrilateral QRST with more illustrations.

Line RT divides the quadrilateral into 2 congruent triangles QRT and SRT. The sum of the angles in each triangle is 180 degrees(98 + 50 + 32)

The area of the quadrilateral = 2 × area of triangle QRT = 2 × area of triangle SRT

Using sine rule,

q/SinQ = t/SinT = r/SinR

24/sin98 = QT/sin50

QT = r = sin50 × 24.24 = 18.57

Also

24/sin98 = QR/sin32

QR = t = sin32 × 24.24 = 12.84

Let us find area of triangle QRT

Area of a triangle

= 1/2 abSinC = 1/2 rtSinQ

Area of triangle QRT

= 1/2 × 18.57 × 12.84Sin98

= 118.06

Therefore, area of quadrilateral QRST = 2 × 118.06 = 236.12

Answer:

  216 square units

Step-by-step explanation:

Apparently, we're supposed to ignore the fact that the given geometry cannot exist. The short diagonal is too short to reach between the angles marked 98°. If Q and S are 98°, then R needs to be 110.13° or more for the diagonals to connect as described.

__

The equal opposite angles of 98° suggests that the figure is symmetrical about the diagonal RT. That being the case, diagonal RT will meet diagonal QS at right angles. Then the area is half the product of the lengths of the diagonals:

  (1/2)×18×24 = 216 . . . . square units

_____

In a quadrilateral, the area can be computed as half the product of the diagonals and the sine of the angle between them. Here, we have assumed the angle to be 90°, so the area is simply half the product of diagonal measures.

Answer:

16 green cubes are in the box.

Step-by-step explanation:

No. of red cubes at first (x):

4(x + 12) = 5(2x)

4x + 48 = 10x

6x = 48

x = 8

No. of green cubes:

= 8 * 2

= 16

Answer: 16 green cubes are in the box.

20 red cubes are in the box in the end.

Proof (the ratio in the end is 4:5):

= 16 is to 20

= 16/4 is to 20/4

= 4 is to 5

Answer:

Option C.

Step-by-step explanation:

A dashed straight line has a negative slope and goes through (0, 2) and (4, 0).

The given inequality is

[tex]y<-\dfrac{1}{2}x+2[/tex]

We need find the point which is a solution to the given linear inequality.

Check the given inequality for point (2, 3).

[tex]3<-\dfrac{1}{2}(2)+2[/tex]

[tex]3<1[/tex]    

This statement is false. Option 1 is incorrect.

Check the given inequality for point (2, 1).

[tex]1<-\dfrac{1}{2}(2)+2[/tex]

[tex]1<1[/tex]

This statement is false. Option 2 is incorrect.

Check the given inequality for point (3, -2).

[tex]-2<-\dfrac{1}{2}(3)+2[/tex]

[tex]-2<0.5[/tex]

This statement is false. Option 3 is correct.

Check the given inequality for point (-1,3).

[tex]3<-\dfrac{1}{2}(1)+2[/tex]

[tex]3<1.5[/tex]

This statement is false. Option 4 is incorrect.

Therefore, the correct option is C.

Answer:

C

Step-by-step explanation:

(2,1)

Answer:

1. B.

2. B.

Step-by-step explanation:

Trigonometry

1) Coterminal angles can be found by adding or subtracting 360° (or 2\pi radians) to a given angle. If we have 120°, adding 360° gives 480°, adding again 360° gives 840°. There is no way to get -180°, so this option is not a coterminal angle to 120°

2)

A. [tex]Cos (-90^o)=0[/tex], and not undefined

B. [tex]Sin (-90^o)=-1[/tex]. This is correct

C.  [tex]Tan (-90^o)[/tex] is undefined, not zero

Thus the only correct option is B.

Answer: $180,000

Step-by-step explanation:

Gross Domestic Product (GDP) is the total monetary value of all finished goods and services made within a country during a specific period. It can be used to estimate the size and growth rate of the country's economy.

In the case above Company X sell leather which is not a finished good to Company Y, so it will not contribute to the gross domestic product (GDP). Company Y sells leather shoes which is a finished good to the consumers, which will contribute to the GDP.

Therefore the total contribution to GDP is $180,000

Answer:

Step-by-step explanation:

The formula for the nth term of a geometric sequence is expressed as follows

Tn = ar^(n - 1)

Where

Tn represents the value of the nth term of the sequence

a represents the first term of the sequence.

n represents the number of terms.

From the information given,

r = 1/3

T3 = - 81

n = 3

Therefore,

- 81 = a× 1/3^(3 - 1)

-81 = a × (1/3)^2

-81 = a/9

a = -81 × 9 = - 729

The exponential equation for this sequence is written as

Tn = - 729 * (1/3)^(n-1)

Therefore, to find the second term,T2, n = 2. It becomes

T2 = - 729 * (1/3)^(2-1)

T2 = - 729 * (1/3)^1

T2 = - 729 * (1/3)

T2 = - 243

Answer:

Step-by-step explanation:

You can identify terms and count them before you start factoring. Doing so will identify 3x² as a term in the numerator, and will show you there are 3 terms in the denominator.

When you factor the expression, you get ...

  [tex]\dfrac{3x^2-3}{3x^2+2x-1}=\dfrac{3(x^2-1)}{(3x-1)(x+1)}=\dfrac{3(x-1)(x+1)}{(3x-1)(x+1)}[/tex]

This reveals a common factor of x+1.

So, the above three observations are true of this rational expression.

Answer:

Market demand at $5 is 12 pork.

Step-by-step explanation:

In a market, the sum of individual demand for a product from buyers is known as market demand.

It is give that the at the price of $3 a pound of pork, Jason buys 8 pounds of pork and Noelle buys 10 pounds of pork.

So, market demand at $3 is

8 + 10 = 18

When the price rises to $5 a pound, Jason buys 5 pounds of pork and Noelle buys 7 pounds of pork.

So, market demand at $5 is

5 + 7 = 12

Therefore, the market demand at $5 is 12 pork.

Answer:

[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]

Step-by-step explanation:

Let's start with the original function.

[tex]y=a sin\frac{2\pi t}{T}[/tex]

We can immediately fill in the amplitude 'a' and period 'T' , as the question defines these for us, and provides values for 'a' and 'T', 4 and 4[tex]\pi[/tex] respectively.

[tex]y=4sin(\frac{2\pi t}{4\pi } )[/tex]

Now we only have phase shift and vertical shift to do. Vertical shift is very easy, you can just add it to the end of the right side of the expression. A positive value will shift the graph up, while a negative value will move shift the graph down. We have '-2' as our value for vertical shift, so we can add that on as so:

[tex]y=4sin(\frac{2\pit }{4\pi } )-2[/tex]

Now phase shift the most complicated of the transformations. Basically, it is just movement left or right. A negative phase shift moves the graph right, a positive phase shift moves the graph left (I know, confusing!). Phase shift applies directly to the x variable, or in this case the t variable. To achieve a -4/3 pi phase shift, we need to input +4/3 pi into the function, because of the aforementioned negative positive rule. Here is what the function looks like with the correct phase shift:

[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]

This function has vertical shift -2, phase shift -4/3 [tex]\pi[/tex], amplitude 4, and period 4[tex]\pi[/tex].

Desmos.com/calculator is a great tool for learning about how various parts of an equation affect the graph of the function, If you want you can input each step of this problem into desmos and watch the graph change to match the criteria.

Answer:

a) d = 143,117 ft

b) h = 612.45 ft

Step-by-step explanation:

If height of the mountain = h  

And distance till the surveyor can see = d = 3200.2 SQRT (h)

Refer to attached file for graphical representation

Then;

A) If h=2000 ft

Then d =3200.2 √ (2000)

d = 3200.2 (44.72)

d = 143,117 ft

B) If d = 15 miles

1mile = 5280 ft

15 mile = 15*5280

15 mile = 79,200 ft

Therefore;

d = 79,200 ft

Since,

d =3200.2 √ (h)  

79,200 = 3200.2 √ (h)

79200/3200.2 =√ (h)

√ (h) = 24.75

{√ (h)} ² = (24.75) ²

h = 612.45 ft

Answer:

Cat  sleeps 420 hours than Alice  in a month

Step-by-step explanation:

Given:

Number of  hours Alice sleeps per night = 9 hours

Number of  hours Cat sleeps per night = 20 hours

To Find:

How many more hours does a cat sleep in 1 month that Alice

Solution:

Let

The total number of hours for which Alice Sleeps in one month be x

The total number of hours for which Cat sleeps in one month be y

Step 1: Number of hours for which Alice Sleeps in one month

X = number of days in a month  X  Number of hours Alice sleeps per night

X = 30 X 9

X = 180 Hours

Step 2: Number of hours for which Cat Sleeps in one month

y= number of days in a month  X  Number of hours cat sleeps per night

y = 30 X 20

y = 600 Hours

Now ,

=> y – x  

=>600 – 180

=>420 hours

Answer:

Step-by-step explanation:

I believe the answer is x² + y² = 9²

Answer:

  3x² +3xh +h²

Step-by-step explanation:

  [tex]\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^3-x^3}{h}=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}{h}\\\\=\dfrac{3x^2h+3xh^2+h^3}{h}=3x^2+3xh+h^2[/tex]

Answer:

It appears to fly faster than its actual speed.

Step-by-step explanation:

In general, if talking about velocities, the direction of the movement should also be taken into account. For example, if two objects move in opposite directions, a person inside one object observes that the other one moving in the opposite goes faster than its actual speed (because the velocities are summed up). If they are moving in the same direction the opposite phenomenon is true (the velocities are subtracted).

Finally, in this example the planes move in opposite direction, therefore, a plane flying in the opposite direction will appear to fly  faster.

Answer:  60 years

Step-by-step explanation:

Let x denotes the age of Demochares .

Time he spent as a boy = [tex]\dfrac{x}{4}[/tex]

Time he spent as a youth = [tex]\dfrac{x}{5}[/tex]

Time he spent as a man= [tex]\dfrac{x}{3}[/tex]

Time he spent in dotage= 13 years

As per given , we have the following equation:

[tex]x=\dfrac{x}{4}+\dfrac{x}{5}+\dfrac{x}{3}+13[/tex]

[tex]x=\dfrac{15x+12x+20x}{60}+13[/tex]  [Take LCM]

[tex]x=\dfrac{47x}{60}+13[/tex]

[tex]x-\dfrac{47x}{60}=13[/tex]

[tex]\dfrac{60x-47x}{60}=13[/tex]

[tex]\dfrac{13x}{60}=13[/tex]

[tex]x=13\times\dfrac{60}{13}=60[/tex]

Hence, he is 60 years old.

The probability of selecting a black card or a jack is 15/26.

Given that, one card is selected from a deck of cards.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favourable outcomes/Total number of outcomes

Total number of outcomes =52

The number of black cards in a deck =26

The number of jack cards in a deck =4

Probability of an event = 26/52 +4/52

= 30/52

= 15/26

Therefore, the probability of selecting a black card or a jack is 15/26.

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Final answer:

The probability of selecting a black card or a jack from a standard deck of 52 cards is 7/13.

Explanation:

To find the probability of selecting a black card or a jack from a standard deck of 52 cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.

In a deck, there are 26 black cards (13 clubs and 13 spades) and a total of 4 jacks.

However, since two of the jacks are black, we must avoid counting them twice.

The probability becomes:

P(Black or Jack) = P(Black) + P(Jack) - P(Black and Jack)

P(Black) = 26/52, P(Jack) = 4/52, and P(Black and Jack) = 2/52

Thus, the probability is:

P(Black or Jack) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13

Therefore, the probability of selecting a black card or a jack from a standard deck is 7/13.

Answer:

P(not yellow or green)=\frac{2}{7}[/tex]

Step-by-step explanation:

a set of cards includes 15 yellow cards, 10 green cards and 10 blue cards

Total cards= 15 yellow + 10 green + 10 blue = 35 cards

Probability of an event = number of outcomes divide by total outcomes

number of outcomes that are not yellow or green are 10 blue cards

So number of outcomes = 10

P(not yellow or green)= [tex]\frac{10}{35} =\frac{2}{7}[/tex]

The probability of choosing a card that is neither yellow nor green from the set is 2/7, as there are 10 blue cards and a total of 35 cards.

The question asks for the probability of choosing a card that is neither yellow nor green from a set containing 15 yellow cards, 10 green cards, and 10 blue cards. To find this probability, we must consider only the blue cards, as they are not yellow or green. The total number of blue cards is 10, and the total number of cards is 35 (since 15 + 10 + 10 = 35).

To calculate the probability, we use the formula:

P(Blue card) = Number of blue cards / Total number of cards = 10 / 35 = 2/7

Thus, the probability of randomly choosing a card that is not yellow or green (i.e., a blue card) is 2/7.

Each person will pay 19.5 dollars.

Step-by-step explanation:

Given

Total bill for dinner = b=$30

First of all we will calculate the 30% of dinner bill to find the amount of tip

So,

[tex]Tip = t = 30\%\ of\ 30\\= 0.30*30\\=9[/tex]

the tip is $9

The total bill including tip will be:

[tex]= 30+9 = \$39[/tex]

Two persons have to divide the tip and dinner equally so,

Each person's share = [tex]\frac{39}{2} = 19.5[/tex]

Hence,

Each person will pay 19.5 dollars.

Keywords: Fractions, division

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Answer:

Step-by-step explanation:

C'(x)=-40+0.08 x

C'(x)=0 gives

-40+0.08 x=0

x=40/0.08=500

C"(x)=0.08>0 at x=500

so C(x) is  minimum if x=500

so 500 cars need to be produced for minimum cost.

or we can solve by completing the squares.

c(x)=12000+0.04(x²-1000 x+250000-250000)

=12000+0.04(x-500)²-0.04×250000

=0.04 (x-500)²+12000-10000

=0.04(x-500)²+2000

c(x) is minimum if x=500

To minimize the cost based on the provided quadratic cost function, 500 cars should be produced.

This a problem of optimization in the arena of Calculus. The cost function C(x) = 12000-40x+0.04x2 is a quadratic function, and the minimum cost occurs at the vertex of the parabola described by this function.  

For any quadratic function f(x)=ax2 +bx + c, minimum or maximum value occurs at x = -b/2a.

In this case, a = 0.04 and b = -40.

So minimum cost occurs when x = -(-40) / 2*0.04 = 500.

So, to incur minimum cost, 500 cars should be produced.

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D) 6x≥360; x≥60

Step-by-step explanation:

The goal is to save at least $350 over the next 6 weeks.

Let the amount to save per week be x

x *6 should be equal or more than the goal.This is

6x ≥ 360

However, dividing the goal amount by number of weeks to get the amount to save per week gives;

360/6 =60

so x≥ 60

The inequality is thus :  6x ≥360;x≥60

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Keywords : goal, save, inequality , weeks

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Final answer:

To complete the statement describing the expression (a+b)(d+e)(a+b)(d+e), we expand it by multiplying each term. Simplifying the expression, we get a²*d² + 2a²de + 2abde + b²e².

Explanation:

To complete the statement describing the expression (a+b)(d+e)(a+b)(d+e), we need to expand it. This can be done by multiplying each term in the first factor by each term in the second factor and then multiplying the result by the third factor. So the expression becomes:

(a+b)(d+e)(a+b)(d+e) = (a*d + a*e + b*d + b*e)(a*d + a*e + b*d + b*e)

We can simplify this further by combining like terms:

(a*d + a*e + b*d + b*e)(a*d + a*e + b*d + b*e) = a²*d² + 2a²de + 2abde + b²e²

Answer:

The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.

Step-by-step explanation:

The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.

We have to find the roots of this given equation.

If a quadratic equation is of the form [tex]ax^{2}+bx +c=0[/tex]

Its roots are [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]

Here the given equation is [tex]2x^{2}-4x-1[/tex] = 0

a = 2

b = -4

c = -1

If the roots are [tex]x_{1} and x_{2}[/tex], then

[tex]x_{1}[/tex] = [tex]\frac{-2+\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]

                       = [tex]\frac{4 +\sqrt{24}}{4}[/tex]

                       = [tex]\frac{2+\sqrt{6} }{2}[/tex]

[tex]x_{2}[/tex] = [tex]\frac{-2-\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]

                        = [tex]\frac{4 +\sqrt{8}}{4}[/tex]

                        = [tex]\frac{2-\sqrt{6} }{2}[/tex]

These are the two roots of the equation.

Hello!

To be quick and simple, your answer would be $0.55

The cost of the chocolate bar in the given scenario is $0.55. This was determined by solving a two-variable system of linear equations from the information provided.

This problem is a classic example of a system of linear equations, specifically two-variable linear equations. Here, we need to find the cost of one chocolate bar and one peanut butter bar, and we have two pieces of information that can be translated into equations. The first information is that a peanut butter bar costs $0.07 more than a chocolate bar. The second is that 5 peanut butter bars and 6 chocolate bars total $6.40. We'll use these equations to solve for the variables.

Let's denote the cost of the chocolate bar as x and the cost of the peanut butter bar as y. Then, from the information given, we can form two equations:

Substitute the first equation into the second to solve for x:

So the cost of the chocolate bar is $0.55.

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Answer:

The imaginary part is 0

Step-by-step explanation:

The number given is:

[tex]x=(\cos(12)+i\sin(12)+ \cos(48)+ i\sin(48))^6[/tex]

First, we can expand this power using the binomial theorem:

[tex](a+b)^k=\sum_{j=0}^{k}\binom{k}{j}a^{k-j}b^{j}[/tex]

After that, we can apply De Moivre's theorem to expand each summand:[tex](\cos(a)+i\sin(a))^k=\cos(ka)+i\sin(ka)[/tex]

The final step is to find the common factor of i in the last expansion. Now:

[tex]x^6=((\cos(12)+i\sin(12))+(\cos(48)+ i\sin(48)))^6[/tex]

[tex]=\binom{6}{0}(\cos(12)+i\sin(12))^6(\cos(48)+ i\sin(48))^0+\binom{6}{1}(\cos(12)+i\sin(12))^5(\cos(48)+ i\sin(48))^1+\binom{6}{2}(\cos(12)+i\sin(12))^4(\cos(48)+ i\sin(48))^2+\binom{6}{3}(\cos(12)+i\sin(12))^3(\cos(48)+ i\sin(48))^3+\binom{6}{4}(\cos(12)+i\sin(12))^2(\cos(48)+ i\sin(48))^4+\binom{6}{5}(\cos(12)+i\sin(12))^1(\cos(48)+ i\sin(48))^5+\binom{6}{6}(\cos(12)+i\sin(12))^0(\cos(48)+ i\sin(48))^6[/tex]

[tex]=(\cos(72)+i\sin(72))+6(\cos(60)+i\sin(60))(\cos(48)+ i\sin(48))+15(\cos(48)+i\sin(48))(\cos(96)+ i\sin(96))+20(\cos(36)+i\sin(36))(\cos(144)+ i\sin(144))+15(\cos(24)+i\sin(24))(\cos(192)+ i\sin(192))+6(\cos(12)+i\sin(12))(\cos(240)+ i\sin(240))+(\cos(288)+ i\sin(288))[/tex]

The last part is to multiply these factors and extract the imaginary part. This computation gives:

[tex]Re x^6=\cos 72+6cos 60\cos 48-6\sin 60\sin 48+15\cos 96\cos 48-15\sin 96\sin 48+20\cos 36\cos 144-20\sin 36\sin 144+15\cos 24\cos 192-15\sin 24\sin 192+6\cos 12\cos 240-6\sin 12\sin 240+\cos 288[/tex]

[tex]Im x^6=\sin 72+6cos 60\sin 48+6\sin 60\cos 48+15\cos 96\sin 48+15\sin 96\cos 48+20\cos 36\sin 144+20\sin 36\cos 144+15\cos 24\sin 192+15\sin 24\cos 192+6\cos 12\sin 240+6\sin 12\cos 240+\sin 288[/tex]

(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)

A calculator simplifies the imaginary part Im(x⁶) to 0