Burt ran the length of the trail in his town’s park. There are 9 parts to the trail, and each part is the same length. Burt ran a total of 1.8 kilometers.

Answer: 4.5×10^0

Step-by-step explanation:

To calculate 9×10²/2×10², you can follow these steps:

Step 1: Simplify the expression by cancelling out the common terms:

9×10²/2×10² = (9/2)×(10²/10²)

Step 2: The 10² terms cancel each other out:

(9/2)×(10²/10²) = (9/2)×1 = 9/2

Step 3: Convert the simplified expression to scientific notation:

9/2 = 4.5, so the expression in scientific notation is 4.5×10^0 (since any number raised to the power of 0 is 1).

The result of the calculation is 4.5×10^0.

Answer:

x = 14°

Step-by-step explanation:

The sum of the angles of a triangle is 180°

.

Let's make an equation according to this:

47° + 103° + (3x - 12)° = 180°

150° + 3x - 12° = 180°

3x = 180° + 12° - 150°

3x = 42° / : 3

x = 14°

The absolute value equation for Kayleigh's location in relation to the high school is |21 - d| = 6. By solving this equation, we find that Kayleigh could live either 15 miles or 27 miles away from the school.

Part A: The absolute value equation would represent the distance between Kayleigh's house and the high school, which can either be closer or further away from the school compared to Donna's house. Given that Donna lives 21 miles from school, and Kayleigh lives 6 miles away from Donna, we can set an equation to indicate all possible locations of Kayleigh's home. The equation would be |21 - d| = 6, where 'd' represents the distance from Kayleigh's house to the high school.

Part B: To understand how far Kayleigh could live from her school, solve the equation above for 'd', you get two solutions: 21 - 6 = 15 miles and 21 + 6 = 27 miles. Therefore, Kayleigh could either live 15 miles or 27 miles away from the school.

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The claim that a line's reflection is actually two parallel lines is false. correct answer is option (A).

A line is reflected by another line, which is the original line's mirror image. Each point on the original line is reflected across a line of reflection, which is often perpendicular to the original line, to create the mirror image. The resulting line is consistent with the original line while being oriented in the other direction.

The statement that a line can be translated into another line with the same length and that it is parallel to the original line is true.

The statement that a line rotates into another line that is the same length and shape as the original line but is orientated at a different angle is also correct.

The rotation of a pair of parallel lines is also valid for a second pair of parallel lines that have the same distance between them but are orientated at a different angle from the initial pair.

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The dollar value, if Ahmed likes 4 times more the chocolate than the vanilla slice, then he finds C four times more valuable than V. Thus, Ahmed placed a chocolate slice is $1.75.

The foundational subject in mathematics, arithmetic covers operations with numbers. They include addition, subtraction, multiplication, and division. One of the major branches of mathematics, arithmetic serves as the cornerstone for students studying the subject of mathematics. Mathematical arithmetic is the study of the characteristics of the conventional operations on numbers.

Using C for the chocolate slice's value and V for the vanilla slice's value

4 slices × C + 4 slices × V = $14

If Ahmed  likes 4 times more the chocolate than the vanilla slice, then he finds C four times more valuable than V, thus

C = 4×V

4 slices ×4V + 4 slices ×V = $24

20 slices ×P = $14

P=$0.7/slice

V= 4×P = 4×$0.7/slice  = $2.8/slice

Thus for a slice that is half chocolate and half vanilla

value= 1/2 slice× C + 1/2 slice × V

= 1/2 slice ( $0.7 /slice + $2.8/slice)

=  $1.75

Hence,  Ahmed placed a chocolate slice is $1.75.

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Answer:

The volume of the prism is 800(sqrt(3)) cubic inches.

Step-by-step explanation:

The volume of a triangular prism can be calculated by multiplying the area of the base (which is an equilateral triangle in this case) by the height of the prism.The area of an equilateral triangle can be calculated using the formula:A = (sqrt(3)/4) x s^2where A is the area and s is the length of one side of the triangle.In this case, the length of one side of the equilateral triangle is 20 inches, so we can substitute that into the formula:A = (sqrt(3)/4) x 20^2

A = (sqrt(3)/4) x 400

A = 100(sqrt(3)) square inchesNow that we have the area of the base, we can calculate the volume of the prism:V = A x h

V = 100(sqrt(3)) x 8

V = 800(sqrt(3)) cubic inches

To find the volume of a triangular prism, one must calculate the area of the base (in this case an equilateral triangle), and then multiply this by the height of the prism. Using the provided dimensions, the volume is calculated to be 800√3 cubic inches.

The question is asking us to find the volume of a triangular prism with an equilateral triangle as the base and a given height. The formula for the volume of a prism is Volume = Base Area * Height. Since the base is an equilateral triangle, its area can be calculated using the formula: Area = (sqrt(3) / 4) * side². By substituting the given side length of 20 inches, we find that the area of the base is sqrt(3) * 100 square inches. We then multiply this by the height of the prism, which is 8 inches, to find the total volume. So, the volume of the prism is 800√3 cubic inches.

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There are 30 elements in region C.

What is the inclusion-exclusion principle?

The inclusion-exclusion principle is a counting technique used in combinatorics to find the size of a set that is a union of several other sets. It states that the size of the union of two or more sets is equal to the sum of their individual sizes minus the size of their intersection(s).

To find n(C), we can use the inclusion-exclusion principle:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

We know that n(U) = 103, so we can use this to find n(B ∩ C) as follows:

n(B ∩ C) = n(B ∪ C) - n(B') = n(U) - n(B' ∩ C') - n(B' ∩ C) - n(B ∩ C') = 103 - 24 - 35 - n(B ∩ C')

Similarly, we can find n(A' ∩ C) and n(A ∩ B' ∩ C) as follows:

n(A' ∩ C) = n(U) - n(A ∪ C') = 103 - n(A) - n(C') + n(A ∩ C') = 103 - 32 - n(C') + 14 = 85 - n(C')

n(A ∩ B' ∩ C) = n(U) - n(A ∪ B' ∪ C') = 103 - n(A ∪ B') - n(C') + n(A ∩ B' ∩ C') = 103 - 69 - n(C') + 6 = 40 - n(C')

Now we can substitute these values into the inclusion-exclusion principle to get:

92 + 52 + n(C) - 11 - 14 - n(B ∩ C) - 32 - (85 - n(C)) - (40 - n(C)) + 6 = n(U)

Simplifying, we get:

n(C) - n(B ∩ C) - 21 = 0

n(C) = n(B ∩ C) + 21

To find n(B ∩ C), we can use the formula we derived earlier:

n(B ∩ C) = 103 - 24 - 35 - n(B ∩ C')

n(B ∩ C') = n(U) - n(B' ∪ C') = n(U) - n(B') - n(C') + n(B' ∩ C') = 103 - 52 - n(C') + 24 = 75 - n(C')

Substituting this into the equation for n(B ∩ C), we get:

n(B ∩ C) = 103 - 24 - 35 - (75 - n(C))

n(B ∩ C) = 39 - n(C)

Now we can substitute this into the equation for n(C) to get:

n(C) = (39 - n(C)) + 21

2n(C) = 60

n(C) = 30

Therefore, there are 30 elements in region C.

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With the given compound interest, the required quarterly payment is $38,745.74.

What is Compound interest?

Compound interest is the interest that is calculated on the initial principal as well as the accumulated interest of previous periods. In other words, it is the interest on interest. The interest is added to the principal at the end of each compounding period, and then the new total amount becomes the principal for the next period's interest calculation.

Now,

The amount financed is equal to the purchase price minus the down payment:

Amount financed = $5,000,000 - 0.15($5,000,000) = $4,250,000

To find the quarterly payment,

PV = Pmt * (1 - (1 + r)⁻ⁿ) / r

where PV is the present value (amount financed), Pmt is the quarterly payment, r is the quarterly interest rate, and n is the total number of quarterly payments (11 years x 4 quarters/year = 44 quarters).

The quarterly interest rate is equal to the annual interest rate divided by 4:

r = 0.091 / 4 = 0.02275

Substituting the values, we get:

$4,250,000 = Pmt * (1 - (1 + 0.02275)⁻⁴⁴) / 0.02275

Solving for Pmt, we get:

Pmt = $38,745.74

Therefore, the required quarterly payment is $38,745.74 (rounded to the nearest cent).

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Answer:

B.

Step-by-step explanation:

I hope this is what you needed. =)

The solution of the exponential equation 3^(3x-2) = 9^(4x-1) is x = 0.

We can solve this exponential equation for x by using logarithms. We can take the logarithm of both sides of the equation, using any base that we prefer. For instance, we can use the natural logarithm, ln:

ln(3^(3x - 2)) = ln(9^(4x - 1))

Now, we can use the properties of logarithms to simplify both sides of the equation. First, recall that ln(a^b) = b ln(a), for any positive value of a and any real value of b. Therefore, we have:

(3x - 2) ln(3) = (4x - 1) ln(9)

Next, we can use another property of logarithms, namely ln(a^b) = b ln(a) = ln(c) → a^b = c, to eliminate the natural logarithms from both sides of the equation. Specifically, we can rewrite ln(9) as ln(3^2), and then use the power rule for logarithms, ln(a^b) = b ln(a), to get:

(3x - 2) ln(3) = (4x - 1) ln(3^2) = 2 (4x - 1) ln(3)

Now, we can simplify the equation by multiplying out the coefficients of ln(3) on the left-hand side:

3x ln(3) - 2 ln(3) = 8x ln(3) - 2 ln(3)

Then, we can collect like terms:

3x ln(3) - 8x ln(3) = -2 ln(3) + 2 ln(3)

Finally, we can solve for x by factoring out ln(3) and dividing both sides by the resulting factor:

(3 ln(3) - 8 ln(3)) x = 0

-5 ln(3) x = 0

x = 0

Therefore, the solution of the exponential equation 3^(3x-2) = 9^(4x-1) is x = 0.

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Answer: No Solutions

Step-by-step explanation:

The square root of -125 if undefined

Therefore, the answer in no solutions

By using the trigonometric identities we get  ∅ = 0.5542.

The six trigonometric ratios serve as the foundation for all trigonometric equations. Sine, cosine, tangent, cosecant, secant, and cotangent are some of their names. The sides of the right triangle, such as the adjacent side, opposite side, and hypotenuse side, are used to describe each of these trigonometric ratios.

The given figure is a right-angled triangle.

We can use trigonometric identities,

Sin∅ =[tex]\frac{adjacent }{hypotenuse}[/tex]

Here we get the value adjacent = 10 and the hypotenuse = 19

Therefore we get the value,

sin∅ = [tex]\frac{10}{19}[/tex] = 0.5263

∅ = [tex]sin^{-1} (0.5263)[/tex]

∅ = 0.5542

Therefore the angle ∅ = 0.5542

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Option D is the correct answer, As a result, LM is 16.1 centimeters long.

The mid-segment of a triangle is aligned to the base and is half the length of the triangle.

MQ = therefore 1/2 LN.

LN = 16.1 centimeters because we know MQ = 8 cm.

MQ is parallel to side LM and half of its length because it is also the midpoint of the triangular LMN.

A line segment linking the midpoints of two of the triangle's sides is known as a mid-segment. It is half the length of the triangle's third edge and parallel to it. In other words, the mid-segment linking the midpoints of sides a and b has length c/2 and is parallel to side c if a triangle has sides of lengths a, b, and c.

Because of this, LM = 2MQ = 16.1 centimeters.

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The situation can be modeled by the following system of linear inequalities:

x ≥ 6 (The number of male volunteers should be at least 6)

x + y ≤ 15 (The total number of volunteers should be no more than 15)

y ≥ 0 (The number of female volunteers should be non-negative)

What is inequality model?

A formula exists for inequality. Less than, larger than, or not equal are used in place of the equal sign when there are inequalities. Rules for resolving disparities are unique. Here are a few examples of inequalities that are mentioned. When disparities are connected, the centre inequality can be jumped over.

The situation can be modeled by the following system of linear inequalities:

x ≥ 6 (The number of male volunteers should be at least 6)

x + y ≤ 15 (The total number of volunteers should be no more than 15)

y ≥ 0 (The number of female volunteers should be non-negative)

Explanation:

The first inequality states that the number of male volunteers, represented by x, should be greater than or equal to 6. This ensures that the science teacher has at least 6 male volunteers.

The second inequality states that the total number of volunteers, represented by x + y, should be no more than 15. This ensures that the science teacher does not exceed the maximum number of volunteers required for the competition.

The third inequality states that the number of female volunteers, represented by y, should be non-negative. This ensures that there are no negative values for the number of female volunteers.

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We can cοnclude that there is a significant difference between the treatments.

This methοd is abοut determining "likely" οr "unlikely" by determining the prοbability assuming the null hypοthesis were true οf οbserving a mοre extreme test statistic in the directiοn οf the alternative hypοthesis than the οne οbserved.

Using the p-value apprοach, we can calculate the p-value fοr the F-statistic.

we see that F = 7.28. The degrees οf freedοm fοr treatments and errοr are df1 = 2 and df2 = 38, respectively. we can find that the p-value is apprοximately 0.0026.

Using the critical-value apprοach, we can find the critical value οf F fοr a significance level οf 0.05 and degrees οf freedοm df1 = 2 and df2 = 38. Frοm a table οr calculatοr, we find that the critical value is apprοximately 3.18.

Since the p-value (0.0026) is less than the significance level (0.05), we reject the null hypοthesis. Alternatively, since the F-statistic (7.28) is greater than the critical value (3.18), we reject the null hypοthesis.

Therefοre, we can cοnclude that there is a significant difference between the treatments.

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The value of x cannot be determined and the answer is undefined.

To find the value of x in this scenario, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, triangle ABC is a right triangle with AB as the hypotenuse. So, we can use the Pythagorean theorem to solve for AC, which is equal to the square root of (AB^2 - BC^2).

Using the given values, we have:
AC = sqrt(16^2 - 8^2)
AC = sqrt(256 - 64)
AC = sqrt(192)
AC ≈ 13.86

Now, we can use triangle ACD to solve for x. This triangle is also a right triangle with CD as the hypotenuse. So, we can use the Pythagorean theorem again to solve for AD, which is equal to the square root of (CD^2 - AC^2).

Using the given values, we have:
AD = sqrt(9^2 - 13.86^2)
AD = sqrt(81 - 192.23)
AD = sqrt(-111.23)

We can't take the square root of a negative number, so there is no real solution for x in this scenario.

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La magnitud física que determina que un trabajador haga el trabajo en un tiempo menor que otro puede ser la RAPIDEZ.

La rapidez o también velocidad (si hablamos de vector) define porque un trabajador realiza un trabajo en 20 segundos y el otro en 12 segundos, significa que el primero es menos rápido que el segundo.  

De cierta manera pudieran estar otras magnitudes relacionadas como la potencia.

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the total tip left by Mary at the coffee shop is $0.11 + $0.99 = $1.10 (rounded to the nearest hundredth).

Why is it?

Let's break down the transactions and calculate the total tip left by Mary.

Mary orders a coffee at $2.15 and tips 5% at the counter:

Cost of coffee = $2.15

Tip = 0.05 x $2.15 = $0.11

Mary orders a latte at $4.50 and doesn't tip:

Cost of latte = $4.50

Mary orders another coffee at $2.15 and leaves a 15% tip for both the latte and coffee:

Cost of coffee = $2.15

Total cost of latte and coffee = $4.50 + $2.15 = $6.65

Tip = 0.15 x $6.65 = $0.99

Therefore, the total tip left by Mary at the coffee shop is $0.11 + $0.99 = $1.10 (rounded to the nearest hundredth).

A tip is an extra amount of money given as a gratuity or a gift to someone for their services, such as for a meal in a restaurant, a hairdresser, or a taxi driver. Tips are typically a percentage of the total cost of the service or goods provided.

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Answer:

13%

Step-by-step explanation:

I have a test on  this  an i'm missing Cake batter. But I have sweet coconut and it says its 13%.

The average annual rate of change during the time period 2007-2012 is approximately 10,797 thousand subscribers per year.

The rate of change function is defined as the rate at which one quantity is changing with respect to another quantity. In simple terms, in the rate of change, the amount of change in one item is divided by the corresponding amount of change in another.

To find the average annual rate of change for this time period, we need to determine the total change in the number of cellular telephone subscribers from 2007 to 2012, and then divide by the number of years in the time period.

The total change in the number of subscribers from 2007 to 2012 is:

335,244 - 270,461 = 64,783

The number of years in the time period is:

2012 - 2007 + 1 = 6

So the average annual rate of change is:

64,783 / 6 = 10,797.17 (rounded to two decimal places)

If we round the average annual rate of change to the nearest unit, it would be 10,797 subscribers per year.

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Tom would bank Ksh 277,272.73 and John would bank Ksh 693,939.39 at the end of the business day with 25% markup.

The markup is the discrepancy between a product's selling price and its cost price. Adding a particular percentage to the cost price to arrive at the selling price of a product is a frequent practise in business. Depending on the sector, the level of competition, and other elements, the markup % may change.

Since it influences a product's or service's profitability, markup is significant in business. If the markup is too low, the company could not have a sufficient profit to pay its bills and continue operating. The product could not be competitive with other similar items on the market if the markup is too high.

Let us suppose the cost price and sales price = x.

Thus,

For Tom:

Cost of 5 A + Cost of 2B = 5x + 2x = 7x

Sale of 10 B = 10(2x) = 20x

Here,

Profit = Sale - Cost = 20x - 7x = 13x

For the given value of profit we have:

Tom's profit = Ksh 230,000

13x = 230,000

x = 17,692.31

Simillarly for Jhon:

or John:

Sale of 3 A = 3(5x) = 15x

Cost of 13 B = 13x

Profit = Sale - Cost = 13x - 15x = -2x

John's profit = Ksh 110,000

-2x = 110,000

x = -55,000

Now, the markup is 25%,:

New cost price for John = 17,692.31

New sale price for John = 17,692.31 + 0.25(17,692.31) = 22,115.39

Actual sale price for John = 22,115.39 - 0.15(22,115.39) = 18,798.08

The new profit is thus:

New cost price for John = 17,692.31

New sale price for John = 17,692.31 + 0.25(17,692.31) = 22,115.39

Actual sale price for John = 22,115.39 - 0.15(22,115.39) = 18,798.08

Hence, Tom would bank Ksh 277,272.73 and John would bank Ksh 693,939.39 at the end of the business day.

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The area of the trapezoid [tex]36^{2}[/tex] centimeters

To use the formula A = ¹h(b₁ + b₂) to find the area of a trapezoid, you need to know the height (h) and the lengths of the two parallel bases (b₁ and b₂).
For example,

The formula for finding the area of a trapezoid is A = ¹h(b₁ + b₂),

Where A represents the area of the trapezoid, h represents the height, and b₁ and b₂ represent the lengths of the two parallel bases.

By plugging in the given values for the height and bases and performing the necessary arithmetic operations, you can find the area of the trapezoid.

It's important to remember to use the correct units and follow the order of operations to obtain the correct result.

If the height of the trapezoid is 9 cm, the lengths of the two bases are 3 cm and 5 cm, you can plug these values into the formula to find the area :
A = ¹h(b₁ + b₂)

= ¹(9)(3 + 5)

= ¹(9)(8)

= 36 square centimeters
So, the area of this trapezoid is 36 square centimeters.
If you have different values for the height and bases, you can use the same formula to find the area.

Just make sure to use the correct units and follow the order of operations (parentheses first, then multiplication and division, then addition and subtraction).

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162,356 doctors were registered that year.

Explanation:

The questions says, "34.8% of all registered doctors are females"

(Consider, total number of registered doctors as [tex]x[/tex])

that's, [tex]34.8\%[/tex] of [tex]x[/tex] are female.

and [tex]34.8\%[/tex] of [tex]x[/tex] is 56,500.

Mathematically,

[tex]\dfrac{34.8}{100} \times x=56500[/tex]

[tex]x=56500\times\dfrac{100}{34.8}[/tex]

[tex]x=162356[/tex] doctors were registered that year.

Hence, the banquet's overall payroll came to $1,495 as the number of captains by their pay rate, and the number of hours worked.

We must first determine the rate of pay for each group of employees before multiplying that number by the total number of employees for the dinner. The pay rate for captains is as follows because they are paid 60% above the rest of the staff: Compensation for captains is $17 per hour plus 60% of that amount, or $17 plus $10.20, for a total hourly wage of $27.20. The pay rate for the remaining employees is: $17 per hour is the pay rate for waitresses, bartenders, and hosts. Now, we can determine the banquet's overall payroll using the formula below.

given

We must first determine the pay rate for each group of employees before multiplying that number by the total number of employees for the dinner.

The pay rate for captains is as follows because they are paid 60% more than the rest of the staff:

Compensation for captains is $17 per hour plus 60% of that amount, or $17 plus $10.20, for a total hourly wage of $27.20.

The pay rate for the remaining employees is:

$17 per hour is the pay rate for servers, bartenders, and hosts.

Now, we can determine the banquet's overall payroll using the formula below:

Total payroll is calculated by multiplying the number of servers by their pay rate, the number of bartenders by their pay rate, the number of hosts by their pay rate, the number of captains by their pay rate, and the number of hours worked.

Total payroll is (14 x $17) plus (5 x $17) plus (4 x $17) plus (2 x $27.20) x 5 = $238 plus $85 plus $68 plus $272 x 5

Total compensation: $1,495

Hence, the banquet's overall payroll came to $1,495 as the number of captains by their pay rate, and the number of hours worked.

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In parabola, the vertex of b(x) is 8 units to the right of the vertex of .

What is parabola in math?

A parabola is a U-shaped plane curve in which every point is situated at an equal distance from both the focus, a fixed point, and the directrix, a fixed line. All of the parabola-related ideas are discussed here since it is a crucial component of the conic section subject. When f(x)=ax² +bx +c, where a, b, and c are real integers and a0, the function is said to be quadratic.

f(x) = x²

b(x) = f(x - 8)

f(x) = x²

f(x- 8) = (x- 8)²

f(X-8) = x² - 16x + 64

b(X) =  f(x- 8) = x² - 16x + 64

Comparing  b(X)  = x² - 16x + 64 with  y = ax² + bx + c

 a = 1, b = -16 , c = 64

-b/2a =  8  ...........1

Hence, the x-coordinate of vertex is 8.

putting x = 8 in b(x)

b(8) = 8² - 16 * 8 + 64 = 0

     So, the vertex of b(x)  is (8,0)

Comparing f(x) = x²  with y = ax² +bx+c

  a = 1 , b= 0 , c = 0

  -b/2a = 0 .........2

 Hence, the x-coordinate of vertex is 0.

putting  x = 0 in f(X)

f(0) = 0² = 0

So, the vertex of  f(x) = (0,0)

Thus, the vertex of b(x) is 8 units to the right of the vertex of .

(Option A is false.)

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The standard form of the expression is 2m² + m.

The standard form is referred to as the general way of representing any type of notation. The standard form formula represents the standard form of an equation which is the commonly accepted form of an equation. For example - The standard form of a polynomial is to write the terms with a higher degree first (descending order of degree) and its coefficients must be in integral form.

Combining like terms, we get:

(m² - m) + (2m + m²) = m² + m² - m + 2m

Simplifying further, we get:

2m² + m

Therefore, the sum of the given expressions is 2m² + m.

To write the answer in standard form, we arrange the terms in descending order of degree of the variable:

2m² + m = 2m² + 1m¹ + 0m⁰

So the standard form of the expression is 2m² + m.

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Answer: Using the cofunction identity for tangent and cotangent:

cot(θ) = 1/tan(θ)

We can rewrite the given equation as:

cot(5θ - 32°) = 1/tan(θ + 26°)

Next, using the identity for the tangent of the sum of two angles:

tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))

We can rewrite the right side of the equation as:

1/tan(θ + 26°) = tan(90° - (θ + 26°)) = tan(64° - θ)

Substituting this back into the original equation:

cot(5θ - 32°) = tan(64° - θ)

Using the identity for the cotangent and tangent of the difference of two angles:

cot(a - b) = (cot(a)cot(b) - 1)/(cot(b) - cot(a))

tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))

We can rewrite the equation as:

(cot(5θ)cot(32°) - 1)/(cot(32°) - cot(5θ)) = (tan(64°)tan(θ) - tan(θ))/(1 + tan(64°)tan(θ))

Simplifying both sides:

(cot(5θ)cot(32°) - 1)/(cot(32°) - cot(5θ)) = (sin(64°)sin(θ))/(cos(64°)cos(θ) + sin(64°)sin(θ))

Cross-multiplying and simplifying:

cos(64°)cos(θ)cot(5θ) - sin(64°)sin(θ)cot(5θ) = -sin(64°)sin(θ)

cos(64°)cos(θ)cot(5θ) = sin(64°)sin(θ)(cot(5θ) + 1)

cos(64°)cos(θ)cot(5θ) = sin(64°)sin(θ)csc(5θ)

cos(64°)cos(θ) = sin(64°)sin(θ)sin(5θ)/cos(5θ)

cos(64°)cos(θ)cos(5θ) = sin(64°)sin(θ)sin(5θ)

Using the identity for the cosine of the sum of two angles:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

We can rewrite the equation as:

cos(64° + θ - 5θ) = 0

cos(64° - 4θ) = 0

64° - 4θ = 90° + k(180°) or 64° - 4θ = 270° + k(180°) where k is an integer

Solving for θ:

64° - 4θ = 90° + k(180°)

-4θ = 26° + k(180°)

θ = -(26°/4) - (k/4)(180°)

θ = -6.5° - 45°k

or

64° - 4θ = 270° + k(180°)

-4θ = 206° + k(180°)

θ = -(206°/4) - (k/4)(180°)

θ = -51.5° - 45°k

Therefore, there are two sets of solutions for θ, given by:

θ = -6.5

Step-by-step explanation: